A THEORY OF BRANCHES FOR ALGEBRAIC CURVES

TRISTRAM DE PIRO

Abstract. This paper develops some of the methods of the ”Italian School” of algebraic geometry in the context of infinitesimals. The results of this paper have no claim to originality, they can be found in [10], we have only made the arguments acceptable by modern standards. However, as the question of rigor was the main criticism of their approach, this is still a useful project. The results are limited to algebraic curves. As well as being interesting in their own right, it is hoped that these may also help the reader to appreciate their sophisticated approach to algebraic surfaces and an understanding of singularities. The constructions are also relevant to current research in Zariski structures, which have played a major role both in model theoretic applications to diophantine geometry and in recent work on non-commutative geometry.

1. Introduction, Preliminary Definitions, Lemmas and Notation

We begin this section with the preliminary reminder to the reader that the following results are concerned with algebraic curves. However, the constructions involved are geometric and rely heavily on the techniques of Zariski structures, originally developed in [12] and [3]. One might, therefore, speculate that the results could, in themselves, be used to develop further the general theory of such structures. Our starting point is the main Theorem 17.1 of [12], also formulated for Zariski geometries in [3];

Theorem 1.1. Main Theorem 17.1 of [12]

Let M be a Zariski structure and C a presmooth Zariski curve in M. If C is non-linear, then there exists a nonconstant continuous map;

f : C → P^{1}(K)

The author was supported by a Marie Curie Research Fellowship.

Moreover, f is a finite map (f^{−1}(x) is finite for every x ∈ C), and for any n, for any definable S ⊆ C^{n}, the image f(S) is a constructible subset (in the sense of algebraic geometry) of [P^{1}(K)]^{n}.

Remarks 1.2. Here, K denotes an algebraically closed field, we refer the reader to the original paper for the remaining terminology of the Theorem.

Using this theorem, one can already see that there exists a close connection between a geometric theory of algebraic curves and Zariski curves. We begin this section by pointing out some of the main obstacles to developing the results of this paper in the context of Zariski curves, leaving the resolution of the main technical problems for another occasion. This discussion continues up to (††), when we introduce the main notation and preliminary lemmas of the paper.

In Section 2, the first major obstacle that we encounter is a suitable generalisation of the notion of a linear system. Using Theorem 1.1, any linear system Σ of algebraic hypersurfaces on [P^{1}(K)]^{n} will define a linear system of Zariski hypersurfaces on C^{n} , by composing with the finite cover f (∗). One would then expect to be able to develop much of the theory of g_{n}^{r} given in Section 2 for such systems, applied to a Zariski curve S ⊆ C^{n}. This follows from the following observations. First, there exists a generalised Bezout’s theorem, holding in Zariski structures, see the paper [12], hence one might hope to obtain an analogous result to Theorem 2.3. Secondly, one would expect that the local calculations, by algebraic power series, which we used in Lemma 2.10, would transfer to intersections on S. This uses the fact that, at a point where f is unramified, S and the algebraic curve f(S) are locally isomorphic, in the sense of infinitesimal neighborhoods, and, at a point p where f is ramified, with multiplicity r, we have the straightforward relation;

I_{italian}^{(Σ)}(p, S, f^{∗}(ϕ_{λ})) = rI_{italian}^{(Σ)}(f(p), f(S), ϕ_{λ})

where I have used the notation of Theorem 2.3, ϕ_{λ} denotes a hypersurface in [P^{1}(K)]^{n} and f^{∗}(ϕ_{λ}) denotes its inverse image in S^{n}.

In Section 3, the notion of a multiple point is introduced. As this is defined locally, one would expect this definition to be generalisable to Zariski curves. Also, the geometric notion of 2 algebraic curves being biunivocal has a natural generalisation to Zariski curves. However, the main result of Section 3, that any algebraic curve is birational(biunivocal) to an algebraic curve without multiple points, is not so easily transferred. This follows from the simple observation that there exist Zariski curves S, which cannot be embedded in a projective space P^{n}(K), see Section 10 of [3]. This failure of Theorem 3.3 could be explained alternatively, by noting that the combinatorics involved must fail for multiplicity calculations using Zariski curves. The solution to this problem seems to be quite difficult and one must presumably attempt to resolve it by introducing a larger class of hypersurfaces than that defined by a linear system Σ, as in (∗) above. See also the remarks below.

In Section 4, the method of Conic projections is not immediately transferable to Zariski structures, as one needs to define what is meant by a line in this context. However, we note the following geometric property of a line in relation to other algebraic curves;

Theorem 1.3. (Luroth) Let f : l → C be a finite morphism of a line l onto a projective algebraic curve C. Then f is biunivocal.

Proof. See [2]. □

This theorem in fact characterises a line l up to birationality. Its proof requires a global topological property of the line, namely that the genus of l is zero. Although the above property may be formulated for Zariski curves, it does not guarantee the existence of such a curve S ⊆ (C^{n})^{eq}, which is not trivially biunivocal to an algebraic line. One would expect the solution of this problem to require more advanced techniques, such as a geometric definition of the genus of a Zariski curve. Severi, in fact, gives such a definition for algebraic curves in [10] and one might hope that his definition would generalise to Zariski curves. One could then hope to extend the methods of Conic projections in this context.

The results of Section 5 rely centrally on the main result of Section 3, hence their generalisation to Zariski curves require a resolution of the problems noted above. One should observe that the Italian geometers definition of a branch is very different to a local definition using algebraic power series, see also the remarks later in this section, hence cannot be straightforwardly generalised using Theorem 1.1. Given a Zariski curve S ⊂ C^{n} and p ∈ S, let {γ_{p}^{1}, …, γ_{p}^{n}} enumerate the branches of f(p) ∈ f(S). We can then define;

\overline{γ}_{p}^{j} = \{x ∈ S ∩ V_{p} : f(x) ∈ γ_{p}^{j}\}

In the case when f is unramified (in the sense of Zariski structures) at p, this would give an adequate definition of a branch \overline{γ}_{p}^{j} for the Zariski curve C. However, the definition is clearly inadequate when p is a point of ramification and requires a generalisation of the methods of Section 5.

The results of Section 6 depend mainly on Cayley’s classification of singularities. As the main technical tool used in the proof of this result is the method of algebraic power series, by using Theorem 1.1, one would expect the results given in this Section to generalise more easily to Zariski curves.

(††) We will work in the language L_{spec}, as defined in [7]. P (L) will denote ⋃_{n≥1} P^{n}(L) where L is an algebraically closed field. Unless otherwise stated, we will assume that the field has characteristic 0, this is to avoid problems concerning Frobenius. The results that we prove in this paper hold in arbitrary characteristic, if we avoid exceptional cases, however we need to make certain modifications to the proof. We will discuss these modifications in the final section of the paper.

We assume the existence of a universal specialisation P (K) →_{π} P (L) where K is algebraically closed and L ⊂ K. Given l ∈ L, we will denote its infinitesimal neighborhood by V_{l}, that is π^{−1}(l). As we noted in the paper [7], it is not strictly necessary to consider a universal specialisation when defining the non-standard intersection multiplicity of curves, one need only consider a prime model of the theory T_{spec}. However, some of our proofs will require more refined infinitesimal arguments which are not first order in the language L_{spec} and therefore cannot immediately be transferred to a prime model. We will only refer to the non-standard model when using infinitesimal arguments.We assume the reader is familiar with the arguments employed in the papers [5], [6] and [12]. Of particular importance are the following notions;

(i). The technique of Zariski structures. We assume that P^{n}(L) may be considered as a Zariski structure in the topology induced by algebraically closed subvarieties. When referring to the dimension of an algebraically closed subvariety V, we will use the model theoretic definition as given in the paper [5]. We will assume the reader is acquainted with the notions surrounding the meaning of ”generic” in this context.

(ii). The method of algebraic power series and their relation to infinitesimals. This technique was explored extensively in the papers [5] and [6]. In the following paper, we will use power series methods to parametrise the branches of algebraic curves without being overly rigorous. By a branch in this context, we refer to the ”Newtonian definition” rather than the one used by the Italian school of algebraic geometry, which is the subject of this paper.

(iii). The non-standard statement and proof of Bezout’s theorem for projective algebraic curves in P^{2}(L). This was given in the paper [6].

We will assume that L has infinite transcendence degree and therefore has the property;

Given any subfield L_{0} ≺ L of finite transcendence degree and an integer n ≥ 1, we can find ā_{n} ∈ P^{n}(L) which is generic over L_{0}. (*)

We refer the reader to (i) above for the relevant definition of generic. In general, when referring to a generic point, we will mean generic with respect to some algebraically closed field of finite transcendence degree. This field will be the algebraic closure of the parameters defining any algebraic object given in the specific context.

By a projective algebraic curve, we will mean a closed irreducible algebraic subvariety C of P^{n}(L) for some n ≥ 1 having dimension 1. Occasionally, we will consider the case when C has distinct irreducible components {C_{1}, …, C_{r}}, which will be made clear in a given situation. In both cases, we need only consider the usual Zariski topology on P^{n}(L) as given in (i) above. By a plane projective algebraic curve, we will mean a projective algebraic curve contained in P^{2}(L). By a projective line l in P^{n}, we will mean a projective algebraic curve isomorphic to P^{1}(L). Any distinct points {p_{1}, p_{2}} in P^{n}(L) determine a unique projective line denoted by l_{p_{1}p_{2}}. We will call the line generic if there exists a generic pair {p_{1}p_{2}} determining it. We occasionally assume the existence of the closed algebraic variety I ⊂ (P^{n} × P^{n}) \ Δ × P^{n}, which parametrises the family of lines in P^{n}, defined by; I(a, b, y) ≡ y ∈ l_{ab}

In order to see that this does define a closed algebraic variety, take the standard open cover {U_{i} := (X_{i} ≠ 0) : 0 ≤ i ≤ n} of P^{n}. Then observe that I∩((U_{i} × U_{j}) \ Δ×P^{n}) is locally trivialisable by the maps;

Θ_{ij} : (U_{i} × U_{j}) \ Δ × P^{1} → I ∩ (U_{i} × U_{j} × P^{n})

Θ_{ij}(a, b, [t_{0} : t_{1}]) =

[t_{0} \frac{a_{0}}{a_{i}} + t_{1} \frac{b_{0}}{b_{j}} : …: t_{0} + t_{1} \frac{b_{i}}{b_{j}} : …: t_{0} \frac{a_{j}}{a_{i}} + t_{1} : …: t_{0} \frac{a_{n}}{a_{i}} + t_{1} \frac{b_{n}}{b_{j}}]

and the transition functions Θ_{ijkl} = Θ_{ij} ∘ Θ_{kl}^{−1} are algebraic. In general, we will leave the reader to check in the course of the paper that certain naturally defined algebraic varieties are in fact algebraic. By a projective plane P in P^{n}, we will mean a closed irreducible projective subvariety of P^{n}, isomorphic to P^{r}(L), for some 0 ≤ r ≤ n. We did, however, use a special notation for a plane projective curve, as defined above. Any sequence of points ā determines a unique projective plane Pā, defined as the intersection of all planes containing ā. We will call a sequence {p_{0}, …, p_{r}} linearly independent if P_{p_{0},...,p_{r}} is isomorphic to P^{r}(L). As before, if U ⊂ (P^{n})^{r+1} defines the open subset of linearly independent elements, we can define the cover I ⊂ U × P^{n}, which parametrises the family of r-dimensional planes in P^{n};

I(p_{0}, …, p_{r}, y) ≡ y ∈ P_{p_{1},...,p_{r}}

As before, one can see that this is a closed projective algebraic subvariety of U × P^{n}.

By a non-singular projective algebraic curve, we mean a projective algebraic curve C ⊂ P^{n} which is non-singular in the sense of [2] (p32). Given any point p ∈ C, we then define its tangent line l_{p} as follows;

By Theorem 8.17 of [2], there exist homogeneous polynomials {G_{1}, …, G_{n−1}} such that C is defined in an affine open neighborhood U of p in P^{n} by the homogenous ideal J =< G_{1}, …, G_{n−1} > (∗). Let dG_{j} = \frac{∂G_{j}}{∂X_{0}} X_{0} +…+ \frac{∂G_{j}}{∂X_{n}} X_{n} be the differential of G_{j}. Then dG_{j} defines a family of hyperplanes, parametrised by an open neighborhood of p in C. If x_{i} = \frac{X_{i}}{X_{0}} is a choice of affine coordinate system containing p and G_{j}^{res}(x_{1}, …, x_{n}) = _{\frac{G_{j}}{X_{0}^{deg(G_{j})}}} , we define dG_{j}^{res} = \frac{∂G_{j}^{res}}{∂x_{1}} dx_{1} + … + \frac{∂G_{j}^{res}}{∂x_{n}} dx_{n}

and J^{res} =< G_{1}^{res}, …, G_{n−1}^{res} >. Then dG_{j}^{res} defines a family of affine hyperplanes, parametrised by an open neighborhood of p in C. We claim that dG_{j}^{res}(p) = dG_{j}(p)^{res}, this follows by an easy algebraic calculation, using the fact that;

\frac{∂G_{j}}{∂X_{0}} (p)p_{0} + … + \frac{∂G_{j}}{∂X_{n}} (p)p_{n} = 0

The differentials {dG_{1}^{res}, …, dG_{n−1}^{res}} are independent at p, by the same Theorem 8.17 of [2], hence ⋂_{1≤j≤n−1} dG_{j}(p), defines a line l_{p} ⊂ P^{n}, which we call the tangent line. We need to show that the definition is independent of the choice of {G_{1}, …, G_{n−1}}. Suppose that we are given another choice {H_{1}, …, H_{n−1}}. Again, using Theorem 8.17 of [2], we can find a matrix (f_{ij})_{1≤i,j≤n−1} of polynomials in R(U), for some affine open neighborhood U of p in P^{n} (∗∗), such that the matrix of values (f_{ij}(p))_{1≤i,j≤n−1} is invertible and;

H_{i}^{res} = ∑_{1≤j≤n−1} f_{ij}G_{j}^{res} (mod (J^{res}(U))^{2})

Then, by properties of differentials, and the fact that, for g ∈ (J^{res}(U))^{2}, dg(p) ≡ 0, we have;

dH_{i}^{res}(p) = ∑_{1≤j≤n−1} f_{ij}(p)dG_{j}^{res}(p)

This implies that ⋂_{1≤j≤n−1} dG_{j}(p) = ⋂_{1≤j≤n−1} dH_{j}(p) as required.

We define the tangent variety T ang(C) to be ⋃_{x∈C} l_{x}. We claim that this is a closed projective subvariety of P^{n}. In order to see this, using the notation of the above argument, let {U_{i}} be an affine cover of P^{n} and let {G_{i1}, …, G_{ij}, …, G_{i,n−1}} be homogeneous polynomials with the properties (∗) and (∗∗) given above. Let W ⊂ P^{n} × P^{n} be the closed projective variety, defined on P^{n} × U_{i} by;

W_{i}(X̄, Ȳ) ≡ U_{i}(Ȳ)∧⋀_{1≤j≤n−1} G_{ij}(Ȳ) = 0∧⋀_{1≤j≤n−1} dG_{ij}(Ȳ)▪X̄ = 0

Then, by completeness, the variety V (X̄) ≡ (∃Ȳ)W (X̄, Ȳ) is a closed projective variety of P^{n}. The above argument shows that V = T ang(C).

We will require a more sophisticated notation when considering hypersurfaces H of P^{n}(L) for n ≥ 1. Namely, by a hypersurface of degree m, we will mean a homogenous polynomial F (X_{0}, X_{1}, …, X_{n}) of degree m in the variables [X_{0} : X_{1} : …: X_{n}]. In the case when F is irreducible or has distinct irreducible factors {F_{1}, …, F_{r}}, such a hypersurface may be considered as the union of r distinct irreducible closed algebraic subvarieties of codimension 1 in P^{n}(L). We will then refer to the hypersurface as having distinct irreducible components. Again, we can understand such hypersurfaces using the usual Zariski topology as given in (i) above. By a hyperplane, we will mean an irreducible hypersurface isomorphic to P^{n−1}(L). Equivalently, a hyperplane is defined by a homogeneous polynomial of degree 1. In general, let F = F_{1}^{n_{1}} ▪ … F_{j}^{n_{j}} ▪ … F_{r}^{n_{r}} be the factorisation of F into irreducibles for 1 ≤ j ≤ r. We will want to take into account the ”non-reduced” character of F if some n_{j} ≥ 2. Therefore, given an irreducible homogeneous polynomial G of degree m and an integer s ≥ 1, we will refer to G^{s} as an s-fold component of degree ms. Geometrically, we interpret G^{s} as follows;

Consider the space of all homogenous polynomials of degree ms parametrised by P^{N}(L). Let W ⊂ P^{N} × P^{n} be the irreducible projective variety defined by W (x̄, ȳ) iff ȳ ∈ Zero(Fx̄), where Fx̄ is the homogenous polynomial (defined uniquely up to scalars) by the parameter x̄. The coefficients of the homogeneous polynomial G^{s} determine uniquely an element ā in P^{N} and we can consider G^{s} as the fibre W (ā) ⊂ P^{n}. In the Zariski topology, this consists of the variety defined by the irreducible polynomial G. However, in the language L_{spec}, we can realise its non-reduced nature using the following lemmas;

Lemma 1.4. Let S ⊂ P^{n} be an irreducible hypersurface and C a projective algebraic curve, then, if C is not contained in S, S ∩ C consists of a finite non-empty set of points.

Proof. The proof follows immediately from the Projective Dimension Theorem, see for example [2]. □

Lemma 1.5. Let G = 0 define an irreducible hypersurface of degree m. Let l be a generic line, then l intersects G = 0 in precisely m points.

Proof. Let P ar_{l} = P^{n} × P^{n} \ Δ be the parameter space for lines in P^{n} as defined above. Let P ar_{m} be the projective parameter space for all homogeneous forms of degree m. Then we can form the variety W ⊂ P ar_{l} × P ar_{m} × P^{n} given by; W (l, x, y) iff y ∈ l ∩ Zero(G_{x}),

where G_{x} is the homogenous polynomial corresponding to the parameter x. If l is chosen to be generic over the parameters defining G_{x}, then l ∩ Zero(G_{x}) is finite. It follows that there exists an open subset U ⊂ P ar_{l} × P ar_{m} consisting of parameters {l, x} such that l ∩ G_{x} has finite intersection. As U is smooth, the finite cover W restricted to U is equidimensional and we may apply Zariski structure arguments, as was done in [6]. Considering W as a finite cover of U, if y ∈ l ∩ Zero(G_{x}), we define Mult_{y}(l, G_{x}) to be Mult_{(l,x,y)}(W/U) in the sense of Zariski structures. Using the notation in [6], we can also define LeftMult_{y}(l, G_{x}) and RightMult_{y}(l, G_{x}). In more geometric language;

LeftMult_{y}(l, G_{x}) = Card(V_{y} ∩ l′ ∩ Zero(G_{x}))

where l′ is a generic infinitesimal variation of l in the nonstandard model P (K).

RightMult_{y}(l, G_{x}) = Card(V_{y} ∩ l ∩ Zero(G_{x′}))

where x′ ∈ V_{x} is generic in P ar_{m}, considered as a variety in the nonstandard model P (K).

We now claim that there exists a line l with exactly m points of intersection with G_{x_{0}} = G. If G_{x_{0}} defines a projective algebraic curve in P^{2}(L), the result follows immediately from an application of Bezout’s theorem and the fact that there exists a line l having only (algebraically) transverse intersections with G_{x_{0}}. This result was shown in [6]. Otherwise, we obtain the case that G_{x_{0}} defines a plane projective curve by repeated application of Bertini’s Theorem for Hyperplane Sections, see [4]. Namely, we can find a 2-dimensional plane P such that P ∩ G_{x_{0}} defines a plane projective algebraic curve C_{x_{0}}. We may assume that such a plane P is determined by a generic triple {a, b, c}. We have an isomorphism θ : P^{2}(L) → P given by;

θ([Y_{0}, Y_{1}, Y_{2}]) = [\frac{a_{0}}{a_{i}} Y_{0} + \frac{b_{0}}{b_{j}} Y_{1} + \frac{c_{0}}{c_{k}} Y_{2} : …: \frac{a_{n}}{a_{i}} Y_{0} + \frac{b_{n}}{b_{j}} Y_{1} + \frac{c_{n}}{c_{k}} Y_{2}]

If the form G_{x_{0}} is given by ∑_{i_{0}+...+i_{n}=m} x_{i_{0}...i_{n}}X_{0}^{i_{0}} … X_{n}^{i_{n}}, then the corresponding curve C_{x_{0}} is given in coordinates {Y_{0}, Y_{1}, Y_{2}} by; C_{x_{0}}(Y_{0}, Y_{1}, Y_{2}) = ∑_{j_{0}+j_{1}+j_{2}=m} F_{j_{0}j_{1}j_{2}}(a, b, c, x̄_{0})Y_{0}^{j_{0}}Y_{1}^{j_{1}}Y_{2}^{j_{2}}

where F_{j_{0}j_{1}j_{2}} is an algebraic function of {a, b, c, x̄} and, in particular, linear in the variables {x̄}. By the assumption that C_{x_{0}} does not vanish identically on P, we must have that F_{j_{0}j_{1}j_{2}} is not identically zero for some (j_{0}j_{1}j_{2}) with j_{0} + j_{1} + j_{2} = m. It follows that C_{x_{0}} defines a plane projective curve of degree m. Now, by the above argument, we may find a line l_{0} ⊂ P having exactly m intersections with C_{x_{0}}. Therefore, l_{0} has exactly m intersections with G_{x_{0}} as well.

Now consider the fibre U(x_{0}) = {l : l ∩ G_{x_{0}} is finite} and restrict the cover W to U(x_{0}). By the above calculation, we have that;

∑_{y∈l_{0}∩G_{x_{0}}} Mult_{l_{0},y}(W/U(x_{0})) = ∑_{y∈l_{0}∩G_{x_{0}}} LeftMult_{y}(l_{0}, G_{x_{0}}) ≥ m (∗)

By elementary properties of Zariski structures, if l is chosen to be generic over the parameters defining G_{x_{0}}, then (∗) holds, with l replacing l_{0}, and, moreover, LeftMult_{y}(l, G_{x_{0}}) = 1 for each intersection y ∈ l ∩ G_{x_{0}}. This implies that l has at least m points of intersection with G_{x_{0}} = G. In order to obtain equality, we now consider the fibre U(l) = {x : l ∩ G_{x} is finite} and restrict the cover W to U(l). It will follow from the result given in the next section, the Hyperspatial Bezout Theorem, that, for any x ∈ U(l);

∑_{y∈l∩G_{x}} Mult_{x,y}(W/U(l)) = ∑_{y∈l∩G_{x}} RightMult_{y}(l, G_{x}) = deg(l)deg(G_{x})

where, for a projective algebraic curve, degree is given by Definition 1.12. We clearly have that deg(l) = 1 and, by assumption, that deg(G_{x_{0}}) = m. Hence, we must have that l intersects G_{x_{0}} in exactly m points and, moreover, RightMult_{y}(l, G_{x_{0}}) = 1 for each intersection y ∈ l ∩ G_{x_{0}} as well.

The lemma is now proved but we can give an algebraic formulation of the result. Namely, if l is determined by the generic pair {a, b}, then we have an isomorphism θ : P^{1}(L) → l given by;

θ([Y_{0}, Y_{1}]) = [\frac{a_{0}}{a_{i}} Y_{0} + \frac{b_{0}}{b_{j}} Y_{1} : …: \frac{a_{n}}{a_{i}} Y_{0} + \frac{b_{n}}{b_{j}} Y_{1}]

If the form G_{x_{0}} is given by ∑_{i_{0}+...+i_{n}=m} x_{i_{0}...i_{n}}X_{0}^{i_{0}} … X_{n}^{i_{n}}, then the equation of G_{x_{0}} on l is given in coordinates {Y_{0}, Y_{1}} by the homogenous polynomial;

P_{x_{0}}(Y_{0}, Y_{1}) = ∑_{j_{0}+j_{1}=m} F_{j_{0}j_{1}}(a, b, x̄_{0})Y_{0}^{j_{0}}Y_{1}^{j_{1}}

We clearly have, by the same reasoning as above, that P_{x_{0}} has degree m. The result of the lemma gives that P_{x_{0}} has exactly m roots in P^{1}(L). Hence, these roots are all distinct. It follows that the scheme theoretic intersection of l and G = 0 consists of m distinct reduced points. That is l intersects G = 0 algebraically transversely.

□

We now extend the lemma to include reducible varieties. As before G is an irreducible algebraic form of degree m and we consider the power G^{s} for some s ≥ 1. We will say that y ∈ l ∩ G^{s} = 0 is counted r-times if RightMult_{y}(l, G^{s}) = r in the sense defined above.

Lemma 1.6. Let G = 0 define an irreducible hypersurface of degree m and let s ≥ 1. Let l be a generic line, then l intersects G^{s} = 0 in m points each counted s-times.

Proof. We first show that l is generic with respect to G = 0. Let λ = {λ_{i}} be the parameters defining G = 0 and let μ = {μ_{j}} be the parameters defining G^{s} = 0. We claim that λ and μ are interdefinable, considered as elements of the projective spaces P ar_{m} and P ar_{ms}, in the structure P (L), which was considered in [5]. The fact that μ ∈ dcl(λ) is clear. Conversely, let α be an automorphism fixing μ and let G_{α(λ)} be the algebraic form of degree m obtained from G = G_{λ}. As (G_{λ})^{s} = G_{μ}, we have that (G_{α(λ)})^{s} = G_{μ}. Hence, Zero(G_{α(λ)}) = Zero(G_{λ}). By the projective Nullstellenstatz, and the fact that both algebraic forms are irreducible, we must have that < G_{λ} >=< G_{α(λ)} >. Hence, there must exist a unit U in the ring L[X_{0}, …, X_{n}] such that G_{λ} = UG_{α(λ)}. As the only such units are scalars, we obtain immediately that α(λ) = λ in P ar_{ms}. As P (L) is sufficiently saturated, we obtain that λ ∈ dcl(μ). By the previous lemma, we obtain that l intersects G = 0 in exactly m points, hence l intersects G^{s} = 0 in exactly m points as well. Therefore, the first part of the lemma is shown. We now apply the Hyperspatial Bezout Theorem, see below, to obtain that;

∑_{y∈l∩G^{s}=0} RightMult_{y}(l, G^{s}) = deg(l)deg(G^{s}) = ms

Hence, the lemma is shown by proving that for any y ∈ l ∩ G^{s} = 0, RightMult_{y}(l, G^{s}) ≥ s. As before, we choose a generic plane P containing l, such that C_{x_{0}} = P ∩ G_{x_{0}} defines a projective algebraic curve of degree m. Using an explicit presentation of an isomorphism θ : P^{2}(L) → L, as was done above, we clearly have that the intersection (P ∩ (G_{x_{0}})^{s} = 0) consists of the non-reduced curve C_{x_{0}}^{s} = 0, which has degree ms. We now apply a result from the paper [6],(Lemma 4.16), which gives that, for y ∈ l ∩ C_{x_{0}};

RightMult_{y}(l, C_{x_{0}}^{s}, P ar_{Q_{ms}}) = s ▪ RightMult_{y}(l, C_{x_{0}}, P ar_{Q_{ms}}) ≥ s

where we have use the fact that the parameter defining C_{x_{0}}^{s} moves in the projective parameter space Q_{ms} defining plane projective curves of degree ms in P^{2}(L). We now claim that, for y ∈ l ∩ G_{x_{0}} = 0;

RightMult_{y}(l, G_{x_{0}}^{s}) ≥ RightMult_{y}(l, C_{x_{0}}^{s}, P ar_{Q_{ms}}) (∗)

Recall that, given G_{x} an algebraic form of degree m, the restriction of G_{x} to the plane P_{abc} is given by the formula;

C_{x}(Y_{0}, Y_{1}, Y_{2}) = ∑_{j_{0}+j_{1}+j_{2}=m} F_{j_{0}j_{1}j_{2}}(a, b, c, x̄)Y_{0}^{j_{0}}Y_{1}^{j_{1}}Y_{2}^{j_{2}}

We first claim that each F_{j_{0}j_{1}j_{2}} is not identically zero (∗∗). Let H_{x} be a hyperplane given in coordinates by;

H_{x} = ∑_{r=0}^{n} x_{i}X_{i} = 0

Then the restriction of H_{x} to P_{abc} is given by the plane;

P_{x} = (∑_{r=0}^{n} \frac{x_{r}a_{r}}{a_{i}})Y_{0} + (∑_{r=0}^{n} \frac{x_{r}b_{r}}{b_{j}})Y_{1} + (∑_{r=0}^{n} \frac{x_{r}c_{r}}{c_{k}})Y_{2}

By elementary linear algebra, we can find hyperplanes {H_{0}, H_{1}, H_{2}} whose restriction to P_{abc} define the planes {Y_{0} = 0, Y_{1} = 0, Y_{2} = 0}. It follows by direct calculation that the algebraic form of degree m defined by H_{0}^{j_{0}}H_{1}^{j_{1}}H_{2}^{j_{2}} restricts to the curve of degree m defined by Y_{0}^{j_{0}}Y_{1}^{j_{1}}Y_{2}^{j_{2}} = 0. Hence, (∗∗) is shown. Now consider the function;

\overline{F} = \{F_{j_{0}j_{1}j_{2}}\} : P ar_{m} → P ar_{Q_{m}}

By earlier remarks, the algebraic function \overline{F} is linear in the variables {x_{i}} and defined over {a, b, c}. Hence, its image defines a plane P ⊂ P ar_{Q_{m}}. By the above calculation, this plane P contains the linearly independent set {p_{j_{0}j_{1}j_{2}} : j_{0} + j_{1} + j_{2} = m} where C_{p_{j_{0}j_{1}j_{2}}} defines the curve Y_{0}^{j_{0}}Y_{1}^{j_{1}}Y_{2}^{j_{2}} = 0. Hence, \overline{F} is surjective. Moreover, by elementary facts about linear maps, the fibres of \overline{F} are equidimensional. We can then show (∗). Suppose that;

RightMult_{y}(l, C_{x_{0}}^{s}, P ar_{Q_{ms}}) = k

Then one can find x′ ∈ V_{x_{1}} ∩ P ar_{Q_{ms}}, generic over the parameter x_{1} defining C_{x_{0}}^{s}, such that C_{x′} intersects l in the distinct points \{y_{1}, …, y_{k}\} ⊂ V_{y}. As x_{1} is regular for the cover \overline{F}, if x_{2} defines G_{x_{0}}^{s}, we can find x′′ ∈ V_{x_{2}} ∩P ar_{ms}, such that \overline{F} (x′′) = x′. The algebraic form defined by G_{x′′} then intersects the plane P_{abc} in the curve C_{x′}, hence it must intersect the line l in the distinct points {y_{1}, …, y_{k}} ⊂ V_{y} as well. This implies that;

RightMult_{y}(l, G_{x_{0}}^{s}) ≥ k

Hence (∗) is shown. The lemma is then proved. In this lemma we have not shown anything interesting algebraically. Namely, if one considers the restriction of G_{x_{0}}^{s} to l, we obtain the homogeneous polynomial P_{x_{0}}^{s}. By the previous lemma, P_{x_{0}} has m distinct roots in P^{1}(L), hence P_{x_{0}}^{s} has m distinct roots with multiplicity s. Therefore, the scheme theoretic intersection of l with G_{x_{0}}^{s} consists of m distinct copies of the non-reduced scheme L[t]/(t)^{s}. The usefulness of the result will be shown in the following lemmas. □

Remarks 1.7. Note that the latter part of the argument in fact shows that, for any line intersecting G_{x_{0}}^{s} in finitely many points, we must have that each point of intersection y is counted at least s-times.

Lemma 1.8. Let F = 0 define a hypersurface of degree k. Let F = F_{1}^{n_{1}} ▪ … ▪ F_{j}^{n_{j}} ▪ … ▪ F_{r}^{n_{r}} be its factorisation into irreducibles, with degree(F_{j}) = m_{j}. Then there exists a line l, intersecting each component F_{j} in exactly m_{j} points, each counted n_{j} times, with the property that the sets {(F_{j} ∩ l) : 1 ≤ j ≤ r} are pairwise disjoint. Moreover, the set of lines having this property form a Zariski open subset of P ar_{l}, defined over the parameters of F = 0.

Proof. Let (x_{1}, …, x_{r}) ∈ P ar_{m_{1}} × … × P ar_{m_{r}} be the tuple defining each reduced irreducible component of F = 0. By an elementary argument, extending the proof in Lemma 1.6, the tuple is interalgebraic with the tuple x ∈ P ar_{k} defining the hypersurface F = 0. Let θ_{(x_{j},m_{j},n_{j})}(y) ⊂ P ar_{l} be the statement that a line l_{y} intersects F_{j} in exactly m_{j} points, each counted n_{j} times;

∃_{z_{1}≠...≠z_{m_{j}}} [⋀_{1≤i≤m_{j}} z_{i} ∈ l_{y} ∩ F_{j,x_{j}} ∧ RightMult_{z_{i}}(l_{y}, F_{j,x_{j}}) = n_{j} ∧ . ∀w(w ∈ l_{y} ∩ F_{j,x_{j}} → ⋁_{1≤i≤m_{j}} z_{i} = w)]

By definability of multiplicity in Zariski structures, Lemmas 1.5 and Lemmas 1.6, each θ_{(x_{j},m_{j},n_{j})}(y) is definable over x_{j} and is a Zariski dense algebraic subset of P ar_{l}. By the previous remark, the complement of θ_{(x_{j},m_{j},n_{j})}(y) in the set of lines having finite intersection with F_{j} is given by;

∃w[w ∈ l_{y} ∩ F_{j,x_{j}} ∧ RightMult_{w}(l_{y}, F_{j,x_{j}}) ≥ n_{j} + 1]

It follows that θ_{(x_{j},m_{j},n_{j})}(y) defines a Zariski open subset of P ar_{l}. Now let;

θ(y) = ⋀_{1≤j≤r} θ_{(x_{j},m_{j},n_{j})}(y)

Then θ(y) defines a Zariski open subset of P ar_{l}. Finally, let W be the the union of the pairwise intersections of the irreducible components F_{j}. Then, by elementary dimension theory, W is Zariski closed of dimension at most n − 2. Hence, the condition on P ar_{l} that a line passes through W defines a proper closed set over the parameters (x_{1}, …, x_{r}). Let U(y) be the Zariski open complement of this set in P ar_{l}. Then any line l satisfying θ(y) ∧ U(y) has the properties required of the lemma.

□

Definition 1.9. We will call a line l satisfying the conclusion of the lemma transverse to F.

We will now give an alternative characterisation of transversality.

Lemma 1.10. Let F = 0 define a hypersurface of degree k. Then a line l is transverse to F iff l intersects F in finitely many points and, for each y ∈ l∩(F = 0), LeftMult_{y}(l, F) = 1. Moreover, the notion of transversality may be formulated by a predicate in the language L_{spec}, T ransverse_{k} ⊂ P ar_{k} × P ar_{l};

T ransverse_{k}(λ, l) iff l is transverse to F_{λ}

Proof. The first part of the proof is clear using previous results of this section. For the second part, use the results in Section 3 of [7]. □

We now have;

Lemma 1.11. A Nullstellensatz for Non-Reduced Hypersurfaces

Let F = 0 define a hypersurface of degree k and let Zero(F) = Zero(F_{λ_{1}}) ∪ … ∪ Zero(F_{λ_{r}}) be its geometric factorisation into irreducibles, using the Zariski topology. Let σ(λ_{j}, m_{j}, n_{j}) ⊂ P ar_{m_{j}}, for 1 ≤ j ≤ r, be the predicates defined in L_{spec} by;

A transverse line to F_{λ_{j}} intersects F_{λ_{j}} in exactly m_{j} points, each counted n_{j} times.

Then the original homogeneous polynomial F is characterised, up to scalars, by the sequence;

(Zero(F_{λ_{1}}), …, Zero(F_{λ_{r}}), σ(λ_{1}, m_{1}, n_{1}), …, σ(λ_{r}, m_{r}, n_{r}))

Proof. By the proof of previous results from this section, the formulae σ(λ_{j}, m_{j}, n_{j}), for 1 ≤ j ≤ r, determine the multiplicity of each component F_{λ_{j}}. The result then follows by uniqueness of factorisation. □

We will refer to a hypersurface as generic if the parameter defining it is generic in the parameter space of all hypersurfaces of the same degree.

Definition 1.12. The degree of a projective algebraic curve C is the number of intersections with a generic hyperplane.

We need to check this is a good definition. Let (P^{n})^{∗} be the dual space of P^{n}. (P^{n})^{∗} is the parameter space for all hyperplanes H in P^{n}. We have that;

{a ∈ (P^{n})^{∗} : dim(C ∩ H_{a}) ≥ 1}

is closed in (P^{n})^{∗}, hence, for generic a ∈ (P^{n})^{∗}, C ∩ H_{a} is finite (and non-empty). Choosing some generic a in (P^{n})^{∗}, let m be the number of intersections of H_{a} with C. Let θ(x) ⊂ (P^{n})^{∗} be the statement;

∃_{x_{1}≠...≠x_{m}}(⋀_{1≤i≤m} x_{i} ∈ C ∩ H_{x} ∧ ∀y(y ∈ C ∩ H_{x} → ⋁_{1≤i≤m} x_{i} = y))

θ(x) is algebraic and defined over ∅, hence, as it contains a, must be Zariski dense in (P^{n})^{∗}. In particular, it contains any generic a in (P^{n})^{∗}.

Definition 1.13. The degree of a hypersurface F is the degree of the homogenous polynomial defining it. (See the above remarks)

In the following paper, the notion of birationality between projective algebraic curves, will be central.

Definition 1.14. We define a linear system Σ on P^{r} to be the collection of algebraic forms of degree k, for some k ≥ 1, corresponding to a plane, which we will denote by P ar_{Σ}, contained in P ar_{k}, the parameter space of homogeneous polynomials of degree k. If P ar_{Σ} has dimension n, we define a basis of Σ to be an ordered set of n + 1 forms corresponding to a maximally independent set of parameters in P ar_{Σ}. Equivalently, a basis of Σ is an ordered system;

{ϕ_{0}(X_{0}, …, X_{r}), …, ϕ_{n}(X_{0}, …, X_{r})}

of homogeneous polynomials of degree k belonging to Σ which are independent, that is there do not exist parameters {λ_{0}, …, λ_{n}} such that;

λ_{0}ϕ_{0} + … + λ_{n}ϕ_{n} ≡ 0

Definition 1.15. Given a linear system Σ of dimension n on P^{r}, we define the base locus of the system Σ by;

Base(Σ) = {x̄ ∈ P^{r} : ϕ_{0}(x̄) = … = ϕ_{n}(x̄) = 0}

for any basis of Σ. Given any 2 bases {ϕ_{0}, …, ϕ_{j}, …, ϕ_{n}} and {ψ_{0}, …, ψ_{i}, …, ψ_{n}} of Σ, we can find an invertible matrix of scalars (λ_{ij})_{0≤i,j≤n} such that;

ψ_{i} = ∑_{j=0}^{n} λ_{ij}ϕ_{j}

Hence, the base locus of Σ is well defined. As a basis corresponds to a maximally independent sequence in P ar_{Σ}, we could equivalently define;

Base(Σ) = {x̄ ∈ P^{r} : ϕ(x̄) = 0}

for every algebraic form ϕ belonging to Σ.

Lemma 1.16. Let Σ be a linear system of dimension n and degree k on P^{r}. Then, a choice of basis B for Σ defines a morphism Φ_{Σ,B} :

P^{r} \ Base(Σ) → P^{n} with the property that Image(ϕ_{Σ,B}) is not contained in any hyperplane section of P^{n}. Moreover, given any 2 bases {B, B′} of Σ, there exists a homography θ_{B,B′} : P^{n} → P^{n} such that;

Φ_{Σ,B′} = θ_{B,B′} ∘ Φ_{Σ,B}

Proof. Given a choice of basis B = {ϕ_{0}, …, ϕ_{n}} for Σ, one checks that the map defined by;

Φ_{Σ,B}([X_{0} : …: X_{r}]) = [ϕ_{0}(X_{0}, …, X_{r}) : …: ϕ_{n}(X_{0}, …, X_{r})]

is a morphism with the required properties. Now suppose that {ϕ_{0}, …, ϕ_{j}, …, ϕ_{n}} and {ψ_{0}, …, ψ_{i}, …, ψ_{n}} are 2 bases B and B′ for Σ. Let (λ_{ij})_{0≤i,j≤n} be the matrix of scalars as given in Definition 1.15. Then one can define a homography θ_{B,B′} by;

θ_{B,B′}([Y_{0} : …: Y_{n}]) = [∑_{j=0}^{n} λ_{0j}Y_{j} : …: ∑_{j=0}^{n} λ_{ij}Y_{j} : …: ∑_{j=0}^{n} λ_{nj}Y_{j}]

It is clear that this homography has the required property of the lemma.

□

Definition 1.17. We define a rational map from P^{r} to P^{n} to be a morphism defined by a choice of basis for a linear system Σ.

Remarks 1.18. Given a linear system Σ, we will generally refer to a morphism given by Lemma 1.16 as simply Φ_{Σ}, leaving the reader to remember that a choice of basis is involved. As any 2 such choices differ by a homography, any properties of one morphism transfer directly to the other, so one hopes that this terminology will not cause confusion. More geometrically, observe that, if x ∈ P^{r} \ Base(Σ), then the set of algebraic forms in Σ, vanishing at x, defines a hyperplane H_{x} ⊂ Σ. A choice of basis {ϕ_{0}, …, ϕ_{n}} for Σ identifies this hyperplane H_{x} with a point [ϕ_{0}(x), …, ϕ_{n}(x)] of the dual space P^{n∗}.

Definition 1.19. We say that two projective algebraic curves C_{1} and C_{2} are birational if there exists U ⊂ C_{1} and V ⊂ C_{2}, with U and V open in C_{1} and C_{2} respectively, such that U and V are isomorphic as algebraic varieties. We will use the notation Φ : C_{1} ↭ C_{2} for a birational map.

We will require the following presentation of birational maps; Lemma 1.20. Let C_{1} ⊂ P^{r} and C_{2} ⊂ P^{n} be birational projective algebraic curves, as in Definition 1.16, with the property that no hyperplane section of P^{r} or P^{n} contains C_{1} or C_{2} respectively. Then we can find linear systems {Σ, Σ′}, rational maps;

ϕ_{Σ} : P^{r} \ Base(Σ) → P^{n} ϕ_{Σ′} : P^{n} \ Base(Σ′) → P^{r}

and open subsets {U′, V′} of {C_{1}, C_{2}}, which are disjoint from {Base(Σ), Base(Σ′)}, such that the restrictions ϕ_{Σ} : U′ → V′ and ϕ_{Σ′} : V′ → U′ are (inverse) isomorphisms.

Proof. As usual, let [X_{0} : …: X_{r}] and [Y_{0} : …: Y_{n}] be homogeneous coordinates for P^{r} and P^{n} respectively. Taking the hyperplanes X_{0} = 0 and Y_{0} = 0, we can find affine presentations;

\frac{L[x_{1},...,x_{r}]}{J_{1}} = R(C_{1} \ C_{1} ∩ (X_{0} = 0)) = R(U′′)

\frac{L[y_{1},...,y_{n}]}{J_{2}} = R(C_{2} \ C_{2} ∩ (Y_{0} = 0)) = R(V′′)

where {U′′, V′′} are open subsets of {C_{1}, C_{2}}, {J_{1}, J_{2}} are prime ideals.

We can then find U′ ⊂ U′′ ∩ U and V′ ⊂ V′′ ∩ V such that U′ and V′ are isomorphic as algebraic subvarieties of C_{1} and C_{2} (consider the elements of U which are mapped to V′′ by the original isomorphism). Now choose polynomials F (x̄) and G(ȳ) such that;

\frac{L[x_{1},...,x_{r}]_{F}}{J′_{1}} = R(U′) \frac{L[y_{1},...,y_{n}]_{G}}{J′_{2}} = R(V′)

As R(U′) ≅ R(V′), we can find rational functions {ϕ_{1}(x̄), …, ϕ_{n}(x̄)} and {ψ_{1}(ȳ), …, ψ_{r}(ȳ)} (with denominators powers of F and G respectively) defining morphisms;

Φ : A^{r} \ {F = 0} → A^{n} Ψ : A^{n} \ {G = 0} → A^{r}

and representing the isomorphism U′ ≅ V′. We now show how to convert Φ into Φ_{Σ}. By equating denominators, we are able to write \{ϕ_{1}(x̄), …, ϕ_{n}(x̄)\} as \{\frac{p_{1}(x̄)}{q(x̄)}, …, \frac{p_{n}(x̄)}{q(x̄)}\}. Now make the substitutions x_{i} = \frac{X_{i}}{X_{0}} in \{p_{1}(x̄), …, p_{n}(x̄), q(x̄)\} and multiply through by the highest power of X_{0} to obtain homogeneous polynomials of the same degree {P_{1}(X̄), …, P_{n}(X̄), Q(X̄)}. Let Σ be the linear system defined by the plane spanned by these homogeneous polynomials and define Φ_{Σ} by;

ρY_{0} = Q(X_{0}, …, X_{r}), ρY_{1} = P_{1}(X_{0}, …, X_{r}), …, ρY_{n} = P_{n}(X_{0}, …, X_{r})

where ρ is a constant of proportionality. (This is an alternative notation for a map between projective spaces, used frequently in papers by the Italian geometers Castelnouvo, Enriques and Severi). By the assumption on {C_{1}, C_{2}} concerning hyperplane sections, the homogeneous polynomials {Q, P_{1}, …, P_{n}} form a basis for Σ, hence this defines a rational map. Similarily, one can find a linear system Σ′ and convert Ψ into a rational map ϕ_{Σ′}. The rest of the properties of the lemma follow immediately from the construction. □

We should also note the following equivalent criteria for birationality of projective algebraic curves;

Lemma 1.21. Let C_{1} and C_{2} be projective algebraic curves. Then C_{1} and C_{2} are birational iff;

(i). There is an isomorphism of function fields L(C_{1}) ≅ L(C_{2}). (ii). (In characteristic 0) There exist a_{1} generic in C_{1}, a_{2} generic in C_{2} and an algebraic relation Rational(x, y) such that a_{1} ∈ dcl_{Rational}(a_{2}) and a_{2} ∈ dcl_{Rational}(a_{1}). . (In charateristic p) One can use the same criteria but must pay attention to the presence of the Frobenius map, see the papers [5] and [8] for details on how to resolve this issue.

Definition 1.22. Let Φ : C_{1} ↭ C_{2} be a birational map, as in Definition 1.19. We define the correspondence Γ_{Φ} ⊂ C_{1} × C_{2} associated to Φ to be;

\overline{(Graph(Φ) ⊂ U × V)}

where, for W an algebraic subset of C_{1} × C_{2}, we let W̄ denote its Zariski closure.

Definition 1.23. Let C_{1} and C_{2} be projective algebraic curves. We will say that 2 birational maps Φ_{1} : C_{1} ↭ C_{2} and Φ_{2} : C_{1} ↭ C_{2} are equivalent if there exists U ⊂ C_{1} such that Φ_{1} and Φ_{2} are both defined and agree on U. Clearly, equivalence of birational maps is an equivalence relation.

Lemma 1.24. Let Φ_{1} : C_{1} ↭ C_{2} and Φ_{2} : C_{1} ↭ C_{2} be equivalent birational maps, then Γ_{Φ_{1}} = Γ_{Φ_{2}}.

Proof. Immediate from the definitions.

□

Definition 1.25. We will denote the equivalence class of a birational map Φ by [Φ]. By the above lemma, we can associate a correspondence Γ_{[Φ]} to an equivalence class of birational maps

Lemma 1.26. Obstruction to Birationality at Singular Points

Let Γ_{[Φ]} be a birational correspondence between C_{1} and C_{2}. If x is a non-singular point of C_{1}, there exists a unique corresponding point y of C_{2} and vice-versa.

Proof. Let U ⊂ C_{1} be the set of non-singular points of C_{1}. We can consider Γ_{Φ} as a cover of U. As U is smooth, we may apply the technique of Zariski structures for this cover. By birationality, if x ∈ U is generic, there exists a unique (xy) ∈ Γ_{Φ} and, moreover, Mult(y/x) = 1. This last fact was shown, for example, in the paper [5] and given originally in [12]. By further properties of Zariski structures, again see either of the above, the total multiplicity of points in the cover over U is preserved, in particular, for any x ∈ U, there exists a unique corresponding (xy) ∈ Γ_{Φ}. □

Remarks 1.27. Note that non-singularity is not necessarily preserved when associating y to x in the above lemma. This motivates the following definition.

Definition 1.28. Given a birational correspondence Γ_{[Φ]}, we define the canonical sets U_{[Φ]} ⊂ Γ_{[Φ]},V_{[Φ]} ⊂ C_{1} and W_{[Φ]} ⊂ C_{2} to be the sets;

U_{[Φ]} = {(x, y) ∈ Γ_{[Φ]} : NonSing(x), NonSing(y)}

V_{[Φ]} = π_{1}(U_{[Φ]})

W_{[Φ]} = π_{2}(U_{[Φ]})

Lemma 1.29. Given a birational correspondence Γ_{[Φ]}, there exists an isomorphism Φ_{1} : V_{[Φ]} → W_{[Φ]} such that Γ_{[Φ]} = Γ_{[Φ_{1}]}

Proof. By an elementary result in algebraic geometry, see for example [2], a morphism Φ : U ⊂ C_{1} → P^{n}, where U is an open subset of C_{1}, extends uniquely to the non-singular points of C_{1}. Combining this with Lemma 1.26, we obtain immediately the result. □

A THEORY OF BRANCHES FOR ALGEBRAIC CURVES 21

We now need to relate the canonical sets of a birational correspondence Γ_{[Φ]} with a particular presentation of [Φ] given by Lemma 1.20; Definition 1.30. Let ϕ_{Σ} be as in Lemma 1.20, we define the canonical open sets associated to ϕ_{Σ} to be;

V_{ϕ_{Σ}} = V_{[Φ]} \ Base(Σ) W_{ϕ_{Σ}} = ϕ_{Σ}(V_{ϕ_{Σ}})

Lemma 1.31. We have the following relations between canonical sets; V_{ϕ_{Σ}} ⊂ V_{[Φ]} ⊂ NonSing(C_{1}) W_{ϕ_{Σ}} ⊂ W_{[Φ]} ⊂ NonSing(C_{2}) ϕ_{Σ} : V_{ϕ_{Σ}} → W_{ϕ_{Σ}} is an isomorphism. V_{ϕ_{Σ}} and Base(Σ) are disjoint.

Proof. The proof is an easy exercise. □ Remarks 1.32. It would be desirable to find a particular presentation Φ_{Σ} of a birational class [Φ], for which Base(Σ) is disjoint from the canonical set V_{[Φ]}. In general, one can easily prove the weaker result;

There exist 2 presentations Φ_{Σ_{1}} and Φ_{Σ_{2}} of a birational class [Φ], such that;

V_{[Φ]} = V_{Φ_{Σ_{1}}} ∪ V_{Φ_{Σ_{2}}} W_{[Φ]} = W_{Φ_{Σ_{1}}} ∪ W_{Φ_{Σ_{2}}} We also note that the choice of Σ presenting a birational class [Φ] is

far from unique. For example, let Id : P^{1} → P^{1} be the identity map. This isomorphism can be represented by any of the birational maps;

ϕ_{n}[X_{0} : X_{1}] = [X_{0}^{n} : X_{0}^{n−1}X_{1}] (n ≥ 1) Let Σ_{n} be the linear system of dimension 1 and degree n defined by

the pair of homogeneous polynomials {X_{0}^{n}, X_{0}^{n−1}X_{1}}. Then ϕ_{n} = Φ_{Σ_{n}} (with this choice of basis).

We finally note the following well known theorem, see for example [2];

Theorem 1.33. Let C be a projective algebraic curve, then C is birational to a plane projective algebraic curve.

2. A Basic Theory of g_{n}^{r}

We begin this section with the definition of intersection multiplicity used by the original Italian school of algebraic geometry.

Definition 2.1. Let C ⊂ P^{w} be a projective algebraic curve. Let P ar_{F} be the projective parameter space for all hypersurfaces of a given degree e and let U = {λ ∈ P ar_{F} : |C ∩ F_{λ}| < ∞} be the open subvariety of P ar_{F} corresponding to hypersurfaces of degree e having finite intersection with C. For λ ∈ U, p ∈ C ∩ F_{λ}, we define;

I_{italian}(p, C, F_{λ}) = Card(C ∩ F_{λ′} ∩ V_{p}) for λ′ ∈ V_{λ} generic in U.

Remarks 2.2. That this is a rigorous definition follows from general properties of Zariski structures. The definition is the same as RightMult_{p}(C, F_{λ}) which we considered in the previous section. We will often abbreviate the notation I_{italian}(p, C, F_{λ}) = s by saying that p is counted s times for the intersection of C with F_{λ}.

The basic theory of g_{n}^{r} relies principally on the following result;

Theorem 2.3. Hyperspatial Bezout

Let C ⊂ P^{w} be a projective algebraic curve of degree d and F_{λ} a hypersurface of degree e having finite intersection with C. Then;

∑_{p∈C∩F_{λ}} I_{italian}(p, C, F_{λ}) = de

We first require the following lemma, preserving the notation from Definition 2.1;

Lemma 2.4. Let H_{λ} be a generic hyperplane, then;

I_{italian}(p, C, H_{λ}) = 1 for all p ∈ C ∩ H_{λ} and each p ∈ C ∩ H_{λ} is non-singular.

In the Italian terminology, each point p of intersection is counted once or the intersection is transverse. Using the methods developed in Section 1, it is not difficult to prove that each point of intersection is transverse (using the scheme theoretic definition).

Proof. Suppose, for contradiction, that I_{italian}(p_{1}, C, H_{λ}) ≥ 2 for some p_{1} ∈ C ∩ H_{λ}. Let {p_{1}, …, p_{d}} be the total set of intersections, where d is the degree of C, see Definition 1.12. Then we can find λ′ ∈ V_{λ} generic in U and a distinct pair {p_{1}^{1}, p_{1}^{2}} in V_{p_{1}} ∩C ∩H_{λ′}. By properties of Zariski structures, see [5] or [12], we can also find {p_{2}^{1}, …, p_{d}^{1}} such that p_{j}^{1} ∈ C ∩ H_{λ′} ∩ V_{p_{j}} for 2 ≤ j ≤ d. It follows from the definition of an infinitesimal neighborhood that, for p ≠ q with {p, q} ⊂ P^{w}, V_{p} and V_{q} are disjoint. Hence, {p_{1}^{1}, p_{1}^{2}, p_{2}^{1}, …, p_{d}^{1}} defines a distinct set of intersections of C with H_{λ′}. It follows that C and H_{λ′} have at least d + 1 intersections, contradicting the facts that λ′ is generic in U and the degree of C is equal to d. For the second part of the lemma, observe that the set of nonsingular points NonSing(C) is a dense open subset of C, defined over the field of definition of C. The condition that a hyperplane passes through at least one point of (C \ NonSing(C)) is therefore a union of finitely many proper hyperplanes P_{1} ∪ … ∪ P_{m} contained in P ar_{H}, also defined over the field of definition of C. As H_{λ} was chosen to be generic, its parameter λ cannot lie inside P_{1}∪…∪P_{m}. Hence, the result is shown.

□

We now complete the proof of Theorem 2.3;

Proof. Choose {λ_{1}, …, λ_{e}} independent generic tuples in P^{w∗} = P ar_{H}, the parameter space for hyperplanes on P^{w}. Let F_{e} be the form of degree e defined by;

F_{e} = H_{λ_{1}} ▪ … ▪ H_{λ_{e}} = Σ_{i_{0}+...+i_{w}=e}λ_{i_{0}...i_{n}}Y_{0}^{i_{0}} … Y_{w}^{i_{w}} = 0

We first claim that the intersections C ∩H_{λ_{j}} are pairwise disjoint sets for 1 ≤ j ≤ e. The condition that a hyperplane H_{μ} passes through at least one point of the intersection {(C∩H_{λ_{1}})∪…∪(C∩H_{λ_{j}})} is a union of finitely many proper closed hyperplane conditions on P ar_{H}, defined over the parameters of C and the tuple {λ_{1}, …, λ_{j}}, for 1 ≤ j ≤ e − 1. As the tuples {λ_{1}, …, λ_{e}} were chosen to be generically independent in P ar_{H}, the result follows. Now, by Lemma 2.4 and the definition of F_{e}, we obtain a total number de of intersections between C and F_{e}. We claim that for each point p of intersection;

I_{italian}(p, C, F_{e}) = 1 (∗)

This does not follow immediately from Lemma 2.4 as the parameter {λ_{i_{0}...i_{n}}} defining F_{e} is allowed to vary in the parameter space P ar_{e} of all forms of degree e. We prove the claim by reducing the problem to one about plane projective curves.

We use Lemma 1.20 and Theorem 1.33 to find a plane projective algebraic curve C_{1} ⊂ P^{2} and a linear system Σ such that Φ_{Σ} : C_{1} ↭ C. Let {ϕ_{0}, …, ϕ_{w}} be a basis for Σ, defining the birational map Φ_{Σ}. We may suppose that each ϕ_{i} is homogenous of degree k in the variables {X_{0}, X_{1}, X_{2}} for P^{2}. Let {V_{[Φ]}, V_{Φ_{Σ}}, W_{[Φ]}, W_{Φ_{Σ}}} be the canonical sets associated to Γ_{Φ_{Σ}}, see Definitions 1.28 and 1.30. Note that the canonical sets are all definable over the data of Φ_{Σ}. Hence, we may, without loss of generality, assume that the point p given in (∗) above lies in W_{Φ_{Σ}} and its corresponding p′ ∈ C_{1} lies in V_{Φ_{Σ}}. In particular, p′ defines a non-singular point of the curve C_{1}. Now, given an algebraic form F_{μ} of degree e;

F_{μ} = Σ_{i_{0}+...+i_{w}=e}μ_{i_{0}...i_{n}}Y_{0}^{i_{0}} … Y_{w}^{i_{w}} = 0

we obtain a corresponding algebraic curve ψ_{μ} of degree ke on P^{2} given by the equation;

ψ_{μ} = Σ_{i_{0}+...+i_{w}=e}μ_{i_{0}...i_{n}}ϕ_{0}^{i_{0}} … ϕ_{w}^{i_{w}} = 0 (†)

We claim that;

I_{italian}(p, C, F_{μ}) ≤ I_{italian}(p′, C_{1}, ψ_{μ}) (∗∗)

For suppose that p is counted s-times for the intersection of C with F_{μ}, then we can find μ′ ∈ V_{μ} generic in U such that C ∩ F_{μ′} ∩ V_{p} consists of the distinct points {p_{1}, …, p_{s}}. By elementary properties of infinitesimals, {p_{1}, …, p_{s}} ⊂ W_{Φ_{Σ}}, hence we can find a corresponding distinct set {p′_{1}, …, p′_{s}} in V_{Φ_{Σ}}. By the defining property of a specialisation ,the fact that the correspondence Γ_{[Φ_{Σ}]} is closed and the definition of ψ_{μ′} we must have that {p′_{1} …, p′_{s}} ⊂ C_{1} ∩ V_{p} ∩ ψ_{μ′}. As the map θ : P ar_{e} → P ar_{ke}, defined by (†), is algebraic, it follows that ψ_{μ′} defines an infinitesimal variation of ψ_{μ} in the space of all algebraic curves of degree ke on P^{2}. Hence, it follows that p′ is counted at least s-times for the intersection of C_{1} with ψ_{μ}. Therefore, (∗∗) is shown. Now, given a hyperplane H_{μ};

H_{μ} = μ_{0}Y_{0} + … + μ_{w}Y_{w} = 0

we obtain a corresponding algebraic curve ϕ_{μ} of degree k on P^{2}, defined by the equation;

ϕ_{μ} = μ_{0}ϕ_{0} + … + μ_{w}ϕ_{w} = 0

Corresponding to the factorisation F_{λ} = F_{e} = H_{λ_{1}} ▪ … ▪ H_{λ_{e}}, we obtain the factorisation ψ_{λ} = ϕ_{λ_{1}} ▪ … ▪ ϕ_{λ_{e}}. Therefore, in order to show (∗), it will be sufficient to prove that;

I_{italian}(p′, C_{1}, ϕ_{λ_{1}} ▪ … ▪ ϕ_{λ_{e}}) = 1 (∗ ∗ ∗)

Let p belong (uniquely) to the intersection C ∩ H_{λ_{j}}. We claim first that;

I_{italian}(p′, C_{1}, ϕ_{λ_{j}}) = 1

We clearly have that the linear system Σ consists of {ϕ_{λ} : λ ∈ P^{w∗}}. Hence, as by construction p′ does not belong to Base(Σ) and is nonsingular, the result in fact follows from Lemma 2.4 and a local result given later in this section, Lemma 2.10, which is independent of this theorem. By results of [6] on plane projective curves, (∗ ∗ ∗) follows. Hence, (∗) is shown as well.

We have now proved that there exists a form F_{e} of degree e which intersects C in exactly de points with multiplicity. The theorem now follows immediately from the corresponding result in Zariski structures that, for a finite equidimensional cover G ⊂ P ar_{e} × P^{w};

Σ_{x∈G(λ)}Mult_{(λ,x)}(G/P ar_{F}) is preserved.

□

Using this theorem, we develop a basic theory of g_{n}^{r} on C, a projective algebraic curve in P^{w}. Suppose that we are given a linear system Σ of dimension r, consisting of algebraic forms ϕ_{λ}, parametrised by P ar_{Σ}. We will assume that C ∩ ϕ_{λ} has finite intersection for each λ ∈ P ar_{Σ}, which we will abbreviate by saying that Σ has finite intersection with C. Then, for λ ∈ P ar_{Σ}, we obtain the weighted set of points;

W_{λ} = {n_{p_{1}}, …, n_{p_{m}}}

where

{p_{1}, …, p_{m}} = C ∩ ϕ_{λ}

and

I_{italian}(p_{j}, C, ϕ_{λ}) = n_{p_{j}} for 1 ≤ j ≤ m.

By Theorem 2.3, the total weight n_{p_{1}} + … + n_{p_{m}} of these points is always equal to de.

It follows that, as λ varies in P ar_{Σ}, we obtain a series of weighted sets Series(Σ) = {W_{λ} : λ ∈ P ar(Σ)}. We now make the following definition;

Definition 2.5. We define order(Series(Σ)) to be the total weight of any of the sets in Series(Σ). We define dim(Series(Σ)) to be dim(Σ). We define g_{n}^{r}(Σ) to be the series of weighted sets parametrised by P ar_{Σ} where;

n = order(Series(Σ)) and r = dim(Series(Σ)).

We now make the following local analysis of g_{n}^{r}(Σ).

Definition 2.6. Let Σ be a linear system having finite intersection with C ⊂ P^{w}. If ϕ_{λ} belongs to Σ and p ∈ C ∩ ϕ_{λ}, we define;

I_{italian}^{Σ}(p, C, ϕ_{λ}) = Card(C ∩ ϕ_{λ′} ∩ V_{p}) for λ′ ∈ V_{λ} generic in P ar_{Σ}.

Remarks 2.7. This is a good definition as P ar_{Σ} is smooth. The difference between I_{italian} and I_{italian}^{Σ} is that in the first case we can vary the parameter λ over all forms of degree e, while, in the second case, we restrict the parameter to forms of the linear system defined by Σ.

Definition 2.8. We will refer to C \ Base(Σ) as the set of mobile points for the system Σ.

We now make the following preliminary definition;

Definition 2.9. We will define a coincident mobile point for Σ to be a point p ∈ C ∩ ϕ_{λ}, for some λ ∈ P ar_{Σ}, such that p lies outside Base(Σ) and with the further property that;

I_{italian}^{Σ}(p, C, ϕ_{λ}) ≨ I_{italian}(p, C, ϕ_{λ})

It is an important property of linear systems in characteristic 0 that there do not exist coincident mobile points. (See the final section for the corresponding result in arbitrary characteristic.) We will prove this in the following lemmas, the notation of Definition 2.6 will be maintained until Lemma 2.17.

Lemma 2.10. Non-Existence of Coincident Mobile Points

Let p ∈ (C \ Base(Σ)) ∩ ϕ_{λ} be a non-singular point. Then;

I_{italian}(p, C, ϕ_{λ}) = I_{italian}^{Σ}(p, C, ϕ_{λ})

Proof. We prove this by induction on m = I_{italian}(p, C, ϕ_{λ}). The case m = 1 is clear as we always have that;

I_{italian}^{Σ}(p, C, ϕ_{λ}) ≤ I_{italian}(p, C, ϕ_{λ})

Suppose that I_{italian}(p, C, ϕ_{λ}) = m+1. Let λ′ ∈ P ar_{Σ}∩V_{λ} be generic and let {p_{1}, …, p_{r}} enumerate V_{p} ∩ C ∩ ϕ_{λ′}. Suppose r ≥ 2, then, by results of [6], (summability of specialisation), we have that;

I_{italian}(p_{j}, C, ϕ_{λ′}) ≤ m for each 1 ≤ j ≤ r.

By properties of infinitesimal neighborhoods, each p_{j} is non-singular and lies in C \ Base(Σ). Hence, by the induction hypothesis, it follows that;

I_{italian}^{Σ}(p_{j}, C, ϕ_{λ′}) = I_{italian}(p_{j}, C, ϕ_{λ′}) for each 1 ≤ j ≤ r

Again, using the same result from [6], (summability of specialisation), we have that; I_{italian}^{(Σ)}(p, C, ϕ_{λ}) = ∑_{1≤j≤r} I_{italian}^{(Σ)}(p_{j}, C, ϕ_{λ′})

where the (Σ) notation is used to show that the result holds for either of the above defined multiplicities. Hence;

I_{italian}(p, C, ϕ_{λ}) = I_{italian}^{Σ}(p, C, ϕ_{λ})

We may, therefore, assume that;

For any generic λ′ ∈ V_{λ}∩P ar_{Σ}, there exists a unique p′ ∈ V_{p}∩C∩ϕ_{λ′}, with I_{italian}(p′, C, ϕ_{λ′}) = I_{italian}(p, C, ϕ_{λ}) (∗)

Now, as p ∉ Base(Σ), we can find ϕ_{μ} with p ∉ C ∩ ϕ_{μ}. We consider the pencil of algebraic forms defined by {ϕ_{λ}, ϕ_{μ}}. We may assume p is in finite position on the curve C. For ease of notation, we will assume that w = 2 and make the generalisation to arbitrary dimension w at the end of the lemma. Let f(X, Y) = 0 define the curve C in affine coordinates such that p corresponds to the point (0, 0). We rewrite the pencil of curves (ϕ_{λ}, ϕ_{μ}) in affine coordinates, which gives the 1parameter family;

g(X, Y; t) = ∑_{i+j≤deg(Σ)}(λ_{ij} + tμ_{ij})X^{i}Y^{j} = 0

The following calculation is somewhat informal, see remark (ii) of Section 1;

As p ∉ Base(Σ), the function \frac{ϕ_{λ}}{ϕ_{μ}} is defined at p = (0, 0) and gives an algebraic morphism;

\frac{ϕ_{λ}}{ϕ_{μ}} : W ⊂ C → P ar_{t}

on some open W ⊂ C, with \frac{ϕ_{λ}}{ϕ_{μ}} (0, 0) = 0.

The cover graph(\frac{ϕ_{λ}}{ϕ_{μ}}) ⊂ W × P ar_{t} of P ar_{t} is Zariski unramified at (0, 0, 0) as, given generic t ∈ V_{0}, there exists a unique (x, y) such that (x, y) ∈ V_{(0,0)} ∩ C ∩ (ϕ_{λ} + tϕ_{μ}), by (∗) and the fact that P ar_{t} is smooth. By results of [8], (Theorem 6.10), if we assume the ground field L has characteristic 0, the cover is etale at (0, 0, 0). We will make the modification for non-zero characteristic in the final section of this paper. (1)

Now, as f(X, Y) = 0 is non-singular at p = (0, 0), we can apply the implicit function theorem to obtain a parametrisation in algebraic power series (x(t), y(t)) of the branch at (0, 0) such that f(x(t), y(t)) = 0. (See the remark (ii) in Section 1 and [6] for the correct interpretation of these power series on appropriate etale covers and the corresponding definition of a branch.) We then obtain a map defined by algebraic power series θ : A^{1,et} → P ar_{t} given by;

θ(t) = \frac{ϕ_{λ}}{ϕ_{μ}} (x(t), y(t))

where A^{1,et} is an etale cover of A^{1} over the distinguished point (0). We have that θ is etale at (0^{lift}, 0), as the composition of etale maps is etale. By the inverse function theorem, we may find an etale isomorphism ρ : A^{1,et} → A^{1,et} such that θ(ρ(t)) = t and (x_{1}(t), y_{1}(t)) = (x(ρ(t)), y(ρ(t))) also parametrises the branch at (0, 0).

We therefore have that;

g(x_{1}(t), y_{1}(t); t) = 0 (**) (2)

Now, by assumption, I_{italian}(p, C, ϕ_{λ}) ≥ 2. Hence, by results of [6];

g(X, Y; 0) is algebraically tangent to f(X, Y) = 0 at (0, 0) (3).

By the chain rule and (∗∗), we have that;

\frac{∂g_{t}}{∂X}_{(x_{1}(t),y_{1}(t))}x′_{1}(t) + \frac{∂g_{t}}{∂Y}_{(x_{1}(t),y_{1}(t))}y′_{1}(t) + \frac{∂g}{∂t}_{(x_{1}(t),y_{1}(t))} = 0 (∗ ∗ ∗) (4)

Hence, at t = 0, we have that \frac{∂g}{∂t}_{(0,0)} = 0, that is p = (0, 0) belongs to ϕ_{μ} = 0, which is a contradiction. The calculation (∗ ∗ ∗) holds for formal power series in L[[t]]. In particular, it holds for algebraic power series. We now give a brief justification for this calculation;

Let v = ord_{t} be the standard valuation on the power series ring L[[t]]. Given a power series f ∈ L[[t]] and a sequence of power series {f_{n} : n ∈ Z_{≥0}}, we will say that {f_{n}} converges to f, abbreviated by {f_{n}} → f, if;

(∀m ∈ Z_{≥0})(∃n(m) ∈ Z_{≥0})(∀k ≥ n(m))[v(f − f_{k}) ≥ m]

Now choose sequences {x_{1}^{n}(t), y_{1}^{n}(t)} of polynomials in L[t] such that {x_{1}^{n}(t)} → x_{1}(t) and {y_{1}^{n}(t)} → y_{1}(t). We claim that;

{g_{n}(t) = g(x_{1}^{n}(t), y_{1}^{n}(t); t)} → g(x_{1}(t), y_{1}(t); t)

This follows by standard continuity arguments for polynomials in the non-archimidean topology induced on L[[t]] by v. We then have that;

g′_{n}(t) = \frac{∂g}{∂X}_{(x_{1}^{n}(t),y_{1}^{n}(t),t)}x_{1}^{n}(t)′ + \frac{∂g}{∂Y} _{(x_{1}^{n}(t),y_{1}^{n}(t),t)}y_{1}^{n}(t)′ + \frac{∂g}{∂t}_{(x_{1}^{n}(t),y_{1}^{n}(t),t)}

This follows from the fact that the chain rule and product rule hold in the polynomial ring L[t], even in non-zero characteristic. We now claim that {x_{1}^{n}(t)′} → x_{1}(t)′ and {y_{1}^{n}(t)′} → y_{1}(t)′. This holds by the definition of convergence and the fact that, for a power series f ∈ L[[t]], if ord_{t}(f) = r, then ord_{t}(f′) ≥ r − 1. Using standard continuity arguments and uniqueness of limits, one obtains the result (4). One can also give a geometric interpretation of the calculation (4) using duality arguments. We will discuss this problem on another occasion.

In order to finish the argument, we claim that;

\frac{∂g_{0}}{∂X}_{(0,0)}x′_{1}(0) + \frac{∂g_{0}}{∂Y}_{(0,0)}x′_{2}(0) = 0 (5)

This follows from (3) and the fact that algebraic tangency can be characterised by the property that Dg_{0} at (0, 0) contains the tangent line l_{p} of C. This is clear if g_{0} is non-singular at p, in particular if g_{0} has a non-reduced component at p. Otherwise, it follows easily from [6] or [2]. Hence, at t = 0, we have that \frac{∂g}{∂t}_{(0,0)} = 0, that is p = (0, 0) belongs to ϕ_{μ} = 0, which is a contradiction.

We now consider the case for arbitrary dimension w. We will use Theorem 1.33 to find a plane projective curve C_{1} ⊂ P^{2} birational to C. Using Lemma 1.20, we can find a linearly independent system Σ′ and a birational presentation Ψ_{Σ′} : C_{1} ↭ C. We will assume that the point p under consideration lies inside the canonical set W_{Ψ_{Σ′}} with corresponding p′ ∈ V_{Ψ_{Σ′}}. This can in fact always be arranged, see the section on Conic Projections. However, for the moment, we can, if necessary, replace the set NonSing(C) by W_{Ψ_{Σ′}}. Now, we follow through the calculation given above for w = 2. The argument up to (1) is unaffected. We first justify the calculation (2). Let f(X, Y) = 0 be an affine representation of C_{1}, such that the point p′ corresponds to (0, 0). Then, we may obtain a local power series representation (x(t), y(t)) of f(X, Y) at (0, 0) and, applying Ψ_{Σ′}, a local power series representation Ψ_{Σ′}(x(t), y(t)) = (x_{1}(t), …, x_{w}(t)) of the corresponding p ∈ C. We may then, applying the same argument, obtain the relation g(x_{1}(t), …, x_{w}(t); t) = 0, where;

g(X_{1}, …, X_{w}; t) = Σ_{i_{1}+...+i_{w}≤deg(Σ)}λ_{i_{1}...i_{w}}X_{1}^{i_{1}} … X_{w}^{i_{w}}+tμ_{i_{1}...i_{w}}X_{1}^{i_{1}} … X_{w}^{i_{w}}

is an affine representation of the pencil ϕ_{λ} + tϕ_{μ}.

We now need to justify the calculation in (3). Write ϕ_{λ} in the form;

ϕ_{λ} = Σ_{i_{0}+...i_{w}=deg(Σ)}λ_{i_{0}...i_{w}}X_{0}^{i_{0}} … X_{n}^{i_{w}} = 0

Let Σ′ = {ψ_{0}, …, ψ_{w}}, then the assumption that ϕ_{λ} passes through p implies that the curve;

D = Σ_{i_{0}+...+i_{w}=deg(Σ)}λ_{i_{0}...i_{w}}ψ_{0}^{i_{0}} … ψ_{i_{w}}^{i_{w}} = 0

passes through the corresponding p′ of C_{1}. By the fact that I_{italian}(p, C, ϕ_{λ}) ≥ 2, we can vary the coefficients {λ_{i_{0}...i_{w}}} of ϕ_{λ} to obtain distinct intersections {x′′, x′′′} in C ∩ ϕ_{λ′} ∩ V_{p}. By properties of infinitesimals, these intersections lie in the fundamental set W_{Ψ_{Σ′}}. Hence, we can find corresponding intersections {x′′′′, x′′′′′} in V_{p′} ∩ V_{ψ_{Σ′}} with the corresponding variation of D. This implies that;

I_{italian}(p′, C_{1}, D) ≥ 2

By results of the paper [6], D must be algebraically tangent to the curve C_{1} at p′. Hence, by the chain rule, and the characterisation of algebraic tangency given above, ϕ_{λ} is algebraically tangent to the curve C at p, in the sense that its differential Dϕ_{λ} at p, contains the tangent line l_{p} of C (∗). The reader should also look at the proof of Theorem 2.3, where a similar calculation was carried out. Finally, we need to justify (4). This is clear from the calculation done above. The final step (5) is also clear from the corresponding calculation and (∗).

□

Remarks 2.11. The lemma fails for non-linear systems. Let C be defined in affine coordinates (x, y) by y = 0 and let {ϕ_{t}} be the pencil of curves, defined in characteristic 0, by y = (x − t)^{2} = x^{2} − 2tx + t^{2}.

By construction, each ϕ_{t} is tangent to y = 0 at (t, 0). It follows that each (t, 0) ∈ C is a coincident mobile point for ϕ_{t}.

Lemma 2.12. Suppose that p ∈ C \ Base(Σ) ∩ ϕ_{λ} is a singular point, then;

I_{italian}(p, C, ϕ_{λ}) = I_{italian}^{Σ}(p, C, ϕ_{λ}).

Proof. Suppose that I_{italian}^{Σ}(p, C, ϕ_{λ}) = m. As p ∉ Base(Σ), the condition that ϕ_{λ} passes through p defines a proper closed subset of the parameter space P ar_{Σ}. Hence, we can find λ′ ∈ V_{λ} generic in P ar_{Σ} and {p_{1}, …, p_{m}} = C ∩ ϕ_{λ′} ∩ V_{p}, distinct from p, witnessing this multiplicity. As both NonSing(C) and C \ Base(Σ) are open and defined over L, we have that {p_{1}, …, p_{m}} must lie in the intersection of these sets. Applying the result of the previous lemma, we must have that;

I_{italian}(p_{j}, C, ϕ_{λ′}) = 1 for 1 ≤ j ≤ m

Hence, by summability of specialisation, again see the paper [6], we must have that I_{italian}(p, C, ϕ_{λ}) = m as required. □

In the following Lemma 2.13, Lemma 2.16 and Lemma 2.17, by a canonical set on C, we will mean either a set of the form V_{ϕ_{Σ_{1}}}, for the domain of a birational map ϕ_{Σ_{1}}, or a set of the form W_{ψ_{Σ_{2}}}, for the image of a birational map ψ_{Σ_{2}}, see also Definition 1.30. For ease of notation, we will abbreviate either of these sets by W. In particular, W may include the canonical set V_{Φ_{Σ}} defined by (any) choice of basis for the linear system Σ.

Lemma 2.13. Multiplicity at non-base points witnessed by transverse intersections in the canonical sets.

Let p ∈ C \ Base(Σ), then, if m = I_{italian}(p, C, ϕ_{λ}), we can find λ′ ∈ V_{λ}, generic in P ar_{Σ}, and distinct {p_{1}, …, p_{m}} = C ∩ϕ_{λ′} ∩V_{p} such that {p_{1}, …, p_{m}} lies in the canonical set W ,with the point p removed, W \ {p}, and the intersection of C with ϕ_{λ′} at each p_{j} is transverse for 1 ≤ j ≤ m.

Proof. As p ∉ Base(Σ), the condition that ϕ_{λ} does not pass through p defines an open subset of P ar_{Σ}. By the previous lemma, taking generic (over L), λ′ ∈ V_{λ}, we can find {p_{1}, …, p_{m}} = C ∩ ϕ_{λ′} ∩ V_{p}, distinct from p. Finally, C \ W defines a finite subset of C (over L). Clearly, {p_{1}, …, p_{m}} avoid this set, otherwise, by properties of specialisations, some p_{j} would equal p for 1 ≤ j ≤ m. Finally, the transversality result follows from the fact that I_{italian}(p_{j}, C, ϕ_{λ′}) = 1 for 1 ≤ j ≤ m, (using Lemmas 2.10 and 2.12 again). □

We have analogous results to Lemmas 2.10, 2.12 and 2.13 for points in Base(Σ);

We first require the following;

Lemma 2.14. Let p ∈ C ∩ Base(Σ), then there exists an open subset U_{p} ⊂ P ar(Σ) and an integer I_{p} ≥ 1 such that;

I_{italian}(p, C, ϕ_{λ}) = I_{p} for λ ∈ U_{p}.

and

I_{italian}(p, C, ϕ_{λ}) ≥ I_{p} for λ ∈ P ar_{Σ}.

Proof. By properties of Zariski structures, we have that;

W_{k} = {λ ∈ P ar(Σ) : I_{italian}(p, C, ϕ_{λ}) ≥ k}

are definable and Zariski closed in P ar_{Σ}. The result then follows by taking I_{p} = min_{λ∈P ar_{Σ}}I_{italian}(p, C, ϕ_{λ}) and the fact that P ar_{Σ} is irreducible.

□

We can now formulate the corresponding version of Lemmas 2.12 and 2.13 for base points;

Lemma 2.15. Let p ∈ C ∩ Base(Σ) ∩ ϕ_{λ}. Then;

I_{italian}(p, C, ϕ_{λ}) = I_{p} + I_{italian}^{Σ}(p, C, ϕ_{λ}) − 1

Proof. Let m = I_{italian}^{Σ}(p, C, ϕ_{λ}). Choosing λ′ ∈ V_{λ} generic in P ar_{Σ}, we can find {p_{1}, …, p_{m−1}} = C ∩ V_{p} ∩ ϕ_{λ′}, distinct from p, witnessing this multiplicity. Therefore, for 1 ≤ j ≤ m − 1, p_{j} ∉ Base(Σ), by properties of specialisations and the fact that Base(Σ) is finite and defined over L. Hence, we can apply the results of Lemmas 2.10 and 2.12 to conclude that I_{italian}(p_{j}, C, ϕ_{λ′}) = 1 for 1 ≤ j ≤ m − 1. As λ′ was generic in P ar_{Σ}, using the previous Lemma 2.14, we have that I_{italian}(p, C, ϕ_{λ′}) = I_{p}. Now, it follows easily, using summability of specialisation (see the paper [6]), that I_{italian}(p, C, ϕ_{λ}) = I_{p} + (m − 1). The lemma is proved. □ Lemma 2.16. Let p ∈ C ∩Base(Σ)∩ϕ_{λ}, then, if m = I_{italian}^{Σ}(p, C, ϕ_{λ}), we can find λ′ ∈ V_{λ}, generic in P ar_{Σ}, and {p, p_{1}, …, p_{m−1}} = C ∩ ϕ_{λ′} ∩ V_{p} witnessing this multiplicity such that {p_{1}, …, p_{m−1}} lie in the canonical set W, see the explanation before Lemma 2.13, and the intersections C ∩ ϕ_{λ′} at p_{j} are transverse for 1 ≤ j ≤ m − 1. Proof. Use the proof of Lemma 2.13, basic properties of infinitesimals and the fact that W is open in C and definable over L. □ Lemma 2.17. Generic Intersections

Fix a canonical set W and let ϕ_{λ} be generic in Σ, then each point of intersection of C with ϕ_{λ} outside Base(Σ) lies inside the canonical set W and is transverse. Proof. The finitely many points of (C \ W) are defined over the data of {W, C}. Hence, the condition on P ar_{Σ} that ϕ_{λ} intersects a point of (C \ W) outside Base(Σ) consists of a finite union of proper hyperplanes defined over the data of {W, C}. Therefore, for generic ϕ_{λ}, each point of intersection of ϕ_{λ} with C, outside Base(Σ), lies inside W. Now observe that the condition of transversality between C and ϕ_{λ}, inside W, defines a constructible condition on P ar_{Σ}, over the data of {C, W }. Namely;

θ(λ) ≡ ∀y[(y ∈ ϕ_{λ} ∩ W) → NonSing(y) ∧ RightMult_{y}(C, ϕ_{λ}) = 1]

By Lemmas 2.13 and 2.16, the condition is Zariski dense in P ar_{Σ}. Hence, the result follows.

□

We now make the following definitions;

Definition 2.18. Let Σ be a linear system defining a g_{n}^{r}(Σ) on a projective algebraic curve C ⊂ P^{w}. Let {W_{λ}} = Series(Σ). If p ∈ W_{λ}, we say that;

p is s-fold (s-plo in Italian) for the g_{n}^{r}(Σ) if I_{italian}(p, C, ϕ_{λ}) ≥ s.

p is counted (contato) s-times for the g_{n}^{r}(Σ) if p has multiplicity s in W_{λ}, equivalently I_{italian}(p, C, ϕ_{λ}) = s.

p has s-fold contact (contatto) with ϕ_{λ} if I_{italian}^{Σ}(p, C, ϕ_{λ}) = s.

Remarks 2.19. The Italian terminology is generally quite confusing. The above lemmas show that the discrepancy between contatto and contato occurs only at fixed points of the system Σ. The philosophy behind their approach is that algebraic calculations may be reduced to visual arguments using the ideas that a s-fold contact at p is a limit of s-points converging along the curve from intersections with forms in the system Σ and that these points are preserved by birationality. In the case of a fixed point, p may be counted more times than its actual contact with ϕ_{λ} and this excess intersection is never actually visually manifested by a variation. The Italian approach is to ignore this excess intersection. This motivates the following definition;

Definition 2.20. Let a g_{n}^{r}(Σ) be given on C. For p ∈ C∩ϕ_{λ}, we define;

I_{italian}^{mobile}(p, C, ϕ_{λ}) = Card(C ∩ϕ_{λ′} ∩{V_{p} \ p}), λ′ ∈ V_{λ} generic in P ar_{Σ}.

Remarks 2.21. One needs to check, as usual, that this is a good definition. This follows, for example, by Lemma 2.14, Remarks 2.7 and Lemma 2.15.

Lemma 2.22. For p ∉ Base(Σ), I_{italian}^{mobile}(p, C, ϕ_{λ}) = I_{italian}(p, C, ϕ_{λ})

For p ∈ Base(Σ), we have that;

I_{p} + I_{italian}^{mobile}(p, C, ϕ_{λ}) = I_{italian}(p, C, ϕ_{λ})

Proof. The proof follows immediately from the same results cited in the previous remark. □

We now make the following definition;

Definition 2.23. By a g_{n}^{r}, we mean the series obtained from a given g_{n′}^{r}(Σ) by removing some (possibly all) of the fixed point contributions I_{x} in Base(Σ). That is, we subtract some part of I_{x} from each weighted set W_{λ}. We define n to be the total multiplicity of each W_{λ} after subtracting some of the fixed point contribution, so n ≤ n′. We say that the g_{n}^{r} has no fixed points if all the fixed point contributions are removed. We refine the Italian terminology for a g_{n}^{r} by saying that x is s-fold for W_{λ} if it appears in the weighted set with multiplicity at least s and x is counted s-times for W_{λ} if it appears with multiplicity exactly s. We define Base(g_{n}^{r}) to be {x ∈ C : ∀λ(x ∈ W_{λ})}.

We now have the following;

Lemma 2.24. For a given g_{n}^{r}, we always have that r ≤ n.

Proof. Let the g_{n}^{r} be defined by Σ, a linear system of dimension r, having finite intersection with C. Pick {p_{1}, …, p_{r}} independent generic points of C, not contained in Base(Σ). The condition that ϕ_{λ} passes through p_{j} defines a proper hyperplane condition H_{p_{j}} on P ar_{Σ}. The base points of the subsystem defined by H_{p_{1}} ∩…∩H_{p_{j}} are defined over p_{1}, …, p_{j} and finite, for 1 ≤ j ≤ r − 1, as, by assumption, no form in Σ contains C. We must, therefore, have that dim(H_{p_{1}} ∩ … ∩ H_{p_{r}}) = 0. That is there exists a unique ϕ_{λ} in Σ passing through {p_{1}, …, p_{r}}. We must have that the total intersection multiplicity of C with ϕ_{λ} outside the fixed contribution is at least r, by construction, hence r ≤ n as required.

□

Definition 2.25. Subordinate Systems

We say that

g_{n_{1}}^{r_{1}} ⊆ g_{n_{2}}^{r_{2}}

if there exist linear systems Σ′ ⊆ Σ (having finite intersection with C) of dimension r_{1} and dimension r_{2} respectively such that g_{n_{1}}^{r_{1}} is obtained from Σ′, g_{n_{2}}^{r_{2}} is obtained from Σ and, for each λ ∈ P ar_{Σ′}, W_{λ} ⊆ V_{λ}. Here, by {W_{λ}, V_{λ}}, we mean the weighted sets parametrised by {g_{n_{1}}^{r_{1}}, g_{n_{2}}^{r_{2}}} and, by W_{λ} ⊆ V_{λ}, we mean that n_{p_{i}} ≤ m_{p_{i}} where W_{λ} = {n_{p_{1}}, …, n_{p_{r}}} and V_{λ} = {m_{p_{1}}, …, m_{p_{r}}}.

Remarks 2.26. Note that the relationship of subordination is clearly transitive. That is, if g_{n_{1}}^{r_{1}} ⊆ g_{n_{2}}^{r_{2}} ⊆ g_{n_{3}}^{r_{3}}, then g_{n_{1}}^{r_{1}} ⊆ g_{n_{3}}^{r_{3}}.

Definition 2.27. Composite Systems

We say that a g_{n}^{r}, defined by Σ, is composite, if, for generic p ∈ C, every weighted set W_{λ} containing p also contains a distinct p′(p) with p′ ∉ Base(Σ).

Remarks 2.28. Note that the definition of composite is well defined, for, if the given g_{n}^{r} is defined by Σ, the statement; θ(x) ≡ ∃x′ ∉ Base(Σ)∀λ ∈ P ar_{Σ}(x ∈ C ∩ ϕ_{λ} → x′ ∈ C ∩ ϕ_{λ})

defines a constructible subset of C. Hence, if it holds for some generic p, it holds for any generic p in C. In modern terminology, we would say that the given g_{n}^{r} seperates points (generically), see Proposition 7.3 of [2].

Definition 2.29. Simple Systems

We say that a g_{n}^{r} is simple if it is not composite.

The importance of simple g_{n}^{r} is due to the following;

Lemma 2.30. Construction of a Birational Model

A simple g_{n}^{r} on C defines a projective image C′ ⊂ P^{r}, birational to C.

Proof. Let the g_{n}^{r} be defined by a linear system Σ, with a choice of basis {ϕ_{0}, …, ϕ_{r}}, (having finite intersection with C), possibly after removing some fixed point contribution. Let Φ_{Σ} be the morphism defined as in Lemma 1.16. This morphism is defined on an open subset U = C \ Base(Σ) of C. By continuity, the image of Φ_{Σ} on U is irreducible, hence either defines a constructible V ⊂ P^{r} of dimension 1 or the image is a point. We can clearly exclude the second case, otherwise we can find {ϕ_{λ}, ϕ_{μ}} differing by a constant of proportionality ρ on U, therefore ϕ_{λ} − ρϕ_{μ} contains C. Let C′ = \overline{V}, then C′ is an irreducible projective algebraic curve. We claim that C′ is birational to C. If not, then, using Lemma 1.21 or just the definition of birationality, in characteristic 0, for generic x ∈ C, we can find a distinct x′ ∈ U such that Φ_{Σ}(x) = Φ_{Σ}(x′). The choice of basis for Σ determines an isomorphism of P ar_{Σ} with P^{r}. Using the parametrisation of P ar_{Σ} given by this isomorphism, if ϕ_{λ} passes through x, the corresponding hyperplane H_{λ} ⊂ P^{r} would pass through Φ_{Σ}(x) and, therefore, ϕ_{λ} would pass through x′ as well. This contradicts simplicity. The lemma may, of course, fail in non-zero characteristic. We refer the reader to the final section for the problems associated to Frobenius. □

We also have the following transfer results;

Lemma 2.31. Let a simple g_{n}^{r} on C be given, as in the previous lemma, defining a birational projective image C′ ⊂ P^{r}. Let V_{ϕ_{Σ}} and W_{ϕ_{Σ}} denote the canonical sets associated to this birational map. Then, given a g_{m}^{d} on C′, without fixed points, there exists a corresponding g_{m}^{d} on C, without fixed points, and, for any corresponding pair {x, x′} in {V_{ϕ_{Σ}}, W_{ϕ_{Σ}}}, x is counted s times in V_{λ} iff x′ is counted s times in W_{λ}, where {V_{λ}, W_{λ}} are the weighted sets parametrised by the g_{m}^{d}, on {C, C′} respectively.

Proof. Let the g_{m}^{d} on C′ be defined by a linear system Σ′ of dimension d, with basis {ψ_{0}, …, ψ_{d}}, after removing all the fixed point contribution. We then obtain a corresponding linear system Σ′′ defined by;

{θ_{λ} = λ_{0}ψ_{0}(ϕ_{0}, …, ϕ_{r}) + … + λ_{d}ψ_{d}(ϕ_{0}, …, ϕ_{r}) = 0}

Clearly, Σ′′ has finite intersection with C, hence it defines a g_{m′}^{d} after removing all fixed point contributions. Now, let {x, x′} be a corresponding pair in {V_{ϕ_{Σ}}, W_{ϕ_{Σ}}}. Suppose that x′ is counted s times in W_{λ}, then, as the g_{m}^{d} on C′ has no fixed points, we have that;

I_{italian}^{mobile}(x′, C′, ψ_{λ}) = s

Therefore, we can find λ′ ∈ V_{λ} generic in P ar_{Σ′} and

{x′_{1}, …, x′_{s}} = C′ ∩ ψ_{λ′} ∩ V_{x′} \ {x′}

By properties of infinitesimals, {x′_{1}, …, x′_{s}} ⊂ W_{ϕ_{Σ}}, hence, we can find corresponding {x_{1}, …, x_{s}} ⊂ V_{ϕ_{Σ}} ∩ θ_{λ′} ∩ V_{x} \ {x}. It follows that I_{italian}^{mobile}(x, C, θ_{λ}) ≥ s, and equality follows from the converse argument. Therefore, as the corresponding g_{m′}^{d} has no fixed points, x is counted s times in V_{λ}. The converse uses the same argument. It remains to show that m = m′. By Lemmas 2.16 and 2.17 and the fact that the g_{m′}^{d} has no fixed points, a generic V_{λ} consists of m′ points each counted once inside V_{ϕ_{Σ}}. Therefore, by the above argument, these points are each counted once inside the corresponding W_{λ}. Hence, m′ ≤ m. We obtain m ≤ m′ by the reversal of this argument.

□

Lemma 2.32. Let a simple g_{n}^{r} on C be given, defining a birational map Φ_{Σ} : C ↭ C′ ⊂ P^{r}. Then, if d is the degree of C′, we have that d′ = d + I for the g_{d′}^{r}(Σ) defined by Σ, where I is the total fixed point contribution from Base(Σ).

Proof. Let k be the degree of ϕ_{Σ} and v the degree of C, then we claim that; d + I = kv

By Lemma 2.4, if H_{λ} is a generic hyperplane, it cuts C′ transversely in d distinct points. We may also assume that these points lie inside the canonical set W_{Φ_{Σ}} as this is defined over the data of Φ_{Σ}. Let {p_{1}, …, p_{d}} be the corresponding points of V_{Φ_{Σ}} and ϕ_{λ} the corresponding form in Σ. By, for example, Lemma 2.17;

I_{italian}(p_{j}, C, ϕ_{λ}) = 1 for 1 ≤ j ≤ d

There can be no more intersections of ϕ_{λ} with C outside Base(Σ), otherwise one would obtain a corresponding intersection of H_{λ} with C′ outside W_{Φ_{Σ}}. Hence, the total multiplicity of intersection I_{italian}(C, ϕ_{λ}) between C and ϕ_{λ} is exactly I + d, using Lemma 2.14. By Theorem 2.3 and the fact that ϕ_{λ} is a form of degree k, we also have that the total multiplicity I_{italian}(C, ϕ_{λ}) is kv. As, by definition, d′ = kv, The result follows.

□

3. The Construction of a Birational Model of a Plane Projective Algebraic Curve without Multiple Points

We first make the following definition.

Definition 3.1. Let C ⊂ P^{w} be a projective algebraic curve, not contained in any hyperplane of P^{w}. We define a point p ∈ C to be s-fold on C if, for every hyperplane H passing through p;

I_{italian}(p, C, H) ≥ s.

and equality is attained.We define p to be a multiple point if it is s-fold for some s ≥ 2. We define p to be simple if it is not multiple.

We have the following lemma;

Lemma 3.2. Let C ⊂ P^{2} be a projective algebraic curve and p a point on C. Let F (X, Y) = 0 be an affine representation of C such that the point p corresponds to (0, 0). Then p is s-fold on C iff we can write F in the form;

F (X, Y) = Σ_{(i+j)≥s}a_{ij}X^{i}Y^{j}

In particular, p is non-singular iff it is not multiple.

Proof. Suppose that p is s-fold on C, then, for a generic line l defined by aX − bY = 0, we have that;

I_{italian}(p, C, l) ≥ s It follows, by the results of [6], that we also have; I_{italian}(p, l, C) = I_{italian}(p, C, l) ≥ s (∗) Now, suppose that F (X, Y) has the expansion; F (X, Y) = Σ_{i+j≤deg(F)}a_{ij}X^{i}Y^{j}. and that a_{ij} ≠ 0 for some (i, j) with i + j < s (**). We can

parametrise the branch at (0, 0) of l algebraically by; (x(t), y(t)) = (bt, at) We make the substitution of this parametrisation into F (X, Y) and

obtain; F (x(t), y(t)) = Σa_{ij}x(t)^{i}y(t)^{j} = Σa_{ij}a^{j}b^{i}t^{i+j} By (∗∗) and generic choice of {a, b}, this expansion has order s_{1} < s.

In characteristic 0, it then follows, by the method of [6], see also Theorem 6.1 of this paper, that, for the pencil Σ_{1} defined by {F_{t} ≡ F (X, Y) + t = 0};

I_{italian}^{Σ_{1}}(p, l, F) = s_{1}. In particular, as p ∉ Base(Σ_{1}), we have, by Lemma 2.10, that; I_{italian}(p, l, F) = s_{1} and therefore, by (∗), that; I_{italian}(p, C, l) = s_{1}

contradicting the fact that p was s-fold on C. It follows that (∗∗) doesn’t hold, as required. See, however, the final section for the problem in non-zero characteristic. For the converse direction, suppose that F (X, Y) has the expansion;

F (X, Y) = Σ_{i+j≥s}a_{ij}X^{i}Y^{j}

Then, by a direct calculation and using the argument above, one has that, for any line;

I_{italian}(p, l, F) ≥ s.

Hence, by (∗) again, p must be s-fold on C.

Using this result, it follows immediately, by considering the vector (\frac{∂F}{∂X} , \frac{∂F}{∂Y}) evaluated at (0, 0), that p is non-singular on C iff p is not multiple.

□

Our main result in this section will be the following;

Theorem 3.3. Let C ⊂ P^{2} be a projective algebraic curve, then there exists C_{1} ⊂ P^{w} such that C and C_{1} are birational and C_{1} has no multiple points.

The proof will proceed using the basic theory of g_{n}^{r} that we developed in the previous section. We require the following definition;

Definition 3.4. Let a g_{n}^{r} without fixed points be given on C. We will call the g_{n}^{r} transverse if the following property holds;

There does not exist a subordinate g_{n′}^{r−1} ⊂ g_{n}^{r} such that n′ ≤ n − 2.

We then have;

Lemma 3.5. Let a g_{n}^{r} on C be given without fixed points. Then, if the g_{n}^{r} is transverse, it must be simple.

Proof. Suppose that the g_{n}^{r}, defined by Σ, is not simple, then it must be composite. Therefore, for generic x ∈ C, there exists an x′(x) ∉ Base(Σ) such that every weighted set W_{λ} containing x also contains x′(x). As x is generic, x ∉ Base(Σ), hence the subsystem Σ_{x} ⊂ Σ, consisting of forms in Σ passing through x, has dimension r −1. Define a g_{n′}^{r−1} from Σ_{x} by removing the fixed point contribution of g_{n′′}^{r−1}(Σ_{x}). We claim that g_{n′}^{r−1} ⊆ g_{n}^{r} (∗). We clearly have that Σ_{x} ⊂ Σ. Suppose that p appears in a W_{λ} defined by g_{n′}^{r−1} with multiplicity s. Then, by definition, I_{italian}^{mobile}(p, C, ϕ_{λ}) = s, calculated with respect to Σ_{x}. It follows easily from the above lemmas that I_{italian}^{mobile}(p, C, ϕ_{λ}) = s, calculated with respect to Σ, as well. Hence, p also appears in the corresponding V_{λ} with multiplicity s. This gives the claim (∗). We show that n′ ≤ n − 2. Let {p_{1}, …, p_{r}} be the base points of Σ. Then Σ_{x} has base points containing the set {p_{1}, …, p_{r}, x, x′(x)}. Clearly, the fixed point contribution of Σ_{x} at {p_{1}, …, p_{r}} can only increase the fixed point contribution of Σ and, moreover, the fixed point contribution of Σ_{x} at {x, x′} must be at least 2. Hence, using the Hyperspatial Bezout Theorem and the definition of {g_{n}^{r}, g_{n′}^{r−1}}, we have that n′ ≤ n−2. This contradicts the fact that the original g_{n}^{r} was transverse. Hence, the g_{n}^{r} must be simple. □

Following from this, we have the following;

Lemma 3.6. Let a g_{n}^{r} be given on C without fixed points such that g_{n}^{r} is transverse. Then the g_{n}^{r} defines a birational map Φ : C ↭ C_{1} ⊂ P^{r} and, moreover, C_{1} has no multiple points.

Proof. The first part of the lemma follows from the previous Lemma 3.5 and Lemma 2.30. See the final section, though, for the problem in non-zero characteristic. It remains to prove that C_{1} has no multiple points. Consider the g_{n′}^{r} without fixed points on C_{1}, defined by the linear system Σ of hyperplane sections. By Lemma 2.31, there exists a corresponding g_{n′}^{r} on C without fixed points. By its construction in Lemma 2.31, it equals the original g_{n}^{r}. Now suppose that there exists a multiple point p of C_{1}. We consider the subsystem Σ_{p} of hyperplane sections passing through p. This defines a g_{n}^{r−1} ⊂ g_{n}^{r} and, as p is multiple, after removing the fixed point contribution, we obtain a g_{n′}^{r−1} ⊂ g_{n}^{r} with n′ ≤ n − 2. Using Lemma 2.31 again, we obtain a corresponding g_{n′}^{r−1} on C. We claim that g_{n′}^{r−1} ⊂ g_{n}^{r} on C. This follows easily from the fact that Σ_{p} ⊂ Σ, the g_{n′}^{r−1} has no fixed points and the argument of the previous lemma. This contradicts the fact that the original g_{n}^{r} on C was transverse. Hence, C_{1} has no multiple points.

□

We now give the proof of Theorem 3.3. We find a transverse g_{n}^{r} on C using a combinatorial argument;

Proof. (Theorem 3.3)

Suppose that C has order n ≥ 2 (the case when C has order n = 1 is obvious.) Consider the independent system Σ consisting of all plane algebraic curves defined by homogeneous forms of order n − 1. Clearly, no form in Σ may contain C, and the system has no fixed points on C, hence the system defines a g_{n_{1}}^{r_{1}}, without fixed points, of dimension r_{1} = \frac{(n−1)(n+2)}{2} and order n_{1} = n(n − 1). If this g_{n_{1}}^{r_{1}} is transverse, the proof is complete. Hence, we may suppose that there exists a g_{n_{2}}^{r_{2}} ⊂ g_{n_{1}}^{r_{1}}, without fixed points, such that r_{2} = r_{1} − 1 and n_{2} ≤ n_{1} − 2. Again, if this g_{n_{2}}^{r_{2}} is transverse, the proof is complete. Hence, we may suppose that there exists a sequence;

g_{n_{i}}^{r_{i}} ⊂ g_{n_{i−1}}^{r_{i−1}} ⊂ … ⊂ g_{n_{1}}^{r_{1}}

with r_{i} = r_{1} − (i − 1) and n_{i} ≤ n_{1} − 2(i − 1). (∗)

We need to show that this sequence terminates. By Lemma 2.24, we always have that r_{i} ≤ n_{i}. Combining this with (∗), we have that;

r_{1} − (i − 1) ≤ n_{1} − 2(i − 1)

Therefore;

i ≤ n_{1} − r_{1} + 1 = n(n − 1) − \frac{(n−1)(n+2)}{2} + 1 = \frac{(n−1)(n−2)}{2} + 1 (∗∗)

Now suppose that equality is attained in (∗∗), then we would have that;

r_{i} = r_{1} − (i − 1) = \frac{(n−1)(n+2)}{2} − \frac{(n−1)(n−2)}{2} = 2(n − 1) ≥ 2

n_{i} ≤ n_{1} − 2(i − 1) = n(n − 1) − (n − 1)(n − 2) = 2(n − 1) (∗ ∗ ∗)

This implies that n_{i} ≤ r_{i} and hence n_{i} = r_{i}. Therefore, we must have that there exists a g_{m}^{m} ⊂ g_{n_{i}}^{r_{i}} for each i with m ≥ 2 and the sequence terminates in fewer than \frac{(n−1)(n−2)}{2} + 1 steps. The final g_{n_{i}}^{r_{i}} in the sequence then defines a birational map from C to C_{1} ⊂ P^{w} without multiple points (w ≥ 2), as required.

□

Remarks 3.7. The terminology of transverse g_{n}^{r} is partly motivated by the following fact. Suppose that a transverse g_{n}^{r} on C is defined by a linear system Σ, possibly after removing some fixed point contribution, then, if x ∈ C \Base(Σ) is non-singular, there exists an algebraic form ϕ_{λ} from Σ such that ϕ_{λ}(x) = 0 but ϕ_{λ} is not algebraically tangent to C at x. The proof of this follows straightforwardly from the definition of transversality and the fact that, if ϕ_{λ} is algebraically tangent to C at x, then I_{italian}(x, C, ϕ_{λ}) ≥ 2 (see the proof of Lemma 4.2). In modern terminology, one calls this property separating tangent vectors. See, for example Proposition 7.3 of [2]. The full motivation, however, comes from Theorem 6.10.

4. The Method of Conic Projections

The purpose of this section is to explore the Italian technique of projecting a curve onto a plane. The method of conic projections is extremely old and can be found in [1], where projective notions are explicitly incorporated in the discussion of perspective. Severi himself also wrote an article on the subject of [1] in [11]. We assume that we are given a projective algebraic curve C ⊂ P^{w} for some w > 2 and that C is not contained in any hyperplane section (otherwise reduce to this lower dimension).

The construction;

Let Ω ⊂ P^{w} be a plane of dimension w − k − 1 and ω ⊂ P^{w} a plane of dimension 2 ≤ k < w such that Ω ∩ ω = ∅. We define the projection of C from Ω to ω as follows;

Let P ∈ C. We may assume that P does not lie on Ω. Let < Ω, P > be the intersection of all hyperplanes containing Ω and P. It is a plane of dimension w − k. Now, by elementary dimension theory, we must have that;

dim(< Ω, P > ∩ω) ≥ k + (w − k) − w = 0

We may exclude the case that < Ω, P > and ω intersect in a line l, as then Ω and l would intersect in a point Q, contradicting the fact that Ω ∩ ω = ∅. Hence, < Ω, P > ∩ω defines a point pr(P). We may repeat this construction for the cofinitely many points U ⊂ C which do not lie on Ω. Now, consider the statement;

ϕ(y) ≡ [y ∈ ω ∧ ∃x∃w(x ∈ Ω ∧ w ∈ U ∧ y ∈ l_{xw})]

By elimination of quantifiers for algebraically closed fields, this clearly defines an algebraic set consisting of {pr(w) : w ∈ U}. We call this the projection pr(U) of U. As U is irreducible, it follows that pr(U) is irreducible. Moreover, pr(U) has dimension 1, otherwise pr(U) would consist of a single point Q, in which case U and therefore C would be contained in the plane < Ω, Q >, contradicting the assumption. We define the projection proj(C) of C from Ω to ω to be the closure \overline{pr(U)} of pr(U) in ω. We can define a correspondence Γ ⊂ U ×pr(U) by;

Γ(w, y) ≡ [y ∈ pr(U) ∧ w ∈ U ∧ ∃x(x ∈ Ω ∧ y ∈ l_{xw})]

We define the associated correspondence Γ_{pr} ⊂ C × proj(C) to be the Zariski closure \overline{Γ}. Note that, in the case when Ω ∩ C = ∅, the correspondence Γ defines an algebraic function (in the sense of model theory) pr from C to pr(C). By the model theoretic description of definable closure in algebraically closed fields of characteristic 0, pr defines a morphism from C to pr(C). See the final section for the problem in non-zero characteristic.

Remarks 4.1. Note that when Ω is a point P and ω is a hyperplane, the construction is equivalent to forming the cone;

Cone(C) = ⋃_{x∈C} l_{xP}

and taking the intersection ω ∩ Cone(C). This is the reason for the terminology of ”conic projections”. The case when w = 3 is explicitly discussed in [1], P represents the eye of the observer wishing to obtain a representation of a curve on a plane.

We now prove the following general lemma on conic projections;

Lemma 4.2. Let w ≥ 3 and suppose that {A, B} are independent generic points of C. Then the line l_{AB} does not otherwise meet the curve.

Proof. Suppose, for contradiction, that we can find independent generic points {A, B} on C such that l_{AB} intersects C in a new point P. As A and B are generic, they are non-singular points on the curve C, hence we can define the tangent lines l_{A} and l_{B} at these points. We claim the following;

For any hyperplane H_{λ} containing l_{A} and passing through A;

I_{italian}(A, C, H_{λ}) ≥ 2 (*)

This is the converse to a result already proved in Lemma 2.10 above. It can be proved in a similar way, namely fix a birational map Φ_{Σ} between C and C_{1} ⊂ P^{2} such that A and its corresponding non-singular point A′ on C_{1} lie in the fundamental sets V_{Φ_{Σ}} and W_{Φ_{Σ}} (use the fact that A is generic). By the chain rule, the corresponding ϕ_{λ} passes through A′ and is tangent to the curve C_{1}. Hence, by results of [6], we have that;

I_{italian}(A′, C_{1}, ϕ_{λ}) ≥ 2

Using the technique of Section 2, it follows easily that (∗) holds as required.

Now choose a generic hyperplane H in P^{w}. Let pr be the projection of C from P to H. Let D = pr(A) = pr(B). We have that pr(C) is defined over P and, moreover, that D is generic in pr(C). This follows as, otherwise, dim(D/P) = 0, therefore, as {A, B} ⊂ l_{P D}, dim(A, B/P) = 0, which implies dim(A, B) = 1, contradicting the fact that {A, B} were independent generic. Hence, D defines a non-singular point on the curve pr(C). Now, let l_{1} and l_{2} be the projections of the lines l_{A} and l_{B}. (We will deal with the degenerate case when one of these projections is a point below (†)). Clearly l_{1} and l_{2} pass through D. We claim that they have the property (∗). Let H_{λ} be a hyperplane of H passing through l_{1}. Then the hyperplane < H_{λ}, P > of P^{w} passes through l_{A}. Let λ′ ∈ V_{λ} be generic in P ar_{H}, then, we may assume that D ∉ pr(C) ∩ H_{λ′}. Therefore, the corresponding hyperplane < H_{λ′}, P > does not pass through A. Now, using the property (∗) of l_{A}, Lemma 2.10 and the fact that < H_{λ′}, P > is an infinitesimal variation of < H_{λ}, P >, we can find distinct {A_{1}, A_{2}} ⊂ V_{A} ∩ C∩ < H_{λ′}, P >. It follows that {pr(A_{1}), pr(A_{2})} ⊂ V_{D} ∩ pr(C) ∩ H_{λ′}. We claim that pr(A_{1}) and pr(A_{2}) are distinct. Suppose not, then pr(A_{1}) = pr(A_{2}) = D′ ∈ V_{D} ∩ pr(C). Consider the following finite cover F ⊂ pr(C) × P^{w};

F (y, x) ≡ y ∈ pr(C) ∧ x ∈ C ∩ l_{yP}

As we have shown, (DA) is generic for this cover, hence, by properties of Zariski structures, mult_{(DA)}(F/pr(C)) = 1. However, this contradicts the fact that we can find D′ ∈ V_{D} ∩ pr(C) and distinct {A_{1}, A_{2}} ⊂ V_{A} such that F (D′A_{1}) and F (D′A_{2}). Therefore, we have shown that I_{italian}(D, pr(C), H_{λ}) ≥ 2. Hence, l_{1} and l_{2} have the property (∗). We now claim that l_{1} = l_{2} (**). Suppose not, then, if l_{D} is the tangent line to pr(C) at D, it must be distinct from say l_{1}. Choose a hyperplane H_{λ} of ω passing passing through l_{1} but not through l_{D}. We have that I_{italian}(D, pr(C), H_{λ}) ≥ 2, but, using part of the proof of Lemma 2.10, H_{λ} must then pass through l_{D}. Hence, (∗∗) is shown.

Now, by (∗∗), we have that the tangent lines l_{A} and l_{B} both lie on the plane spanned by l_{B} and l_{AB}. Consider the statement;

x ∈ NonSing(C) and the tangent line l_{x} lies on the plane spanned by l_{B} and l_{xB}.

It defines an algebraic subset of NonSing(C) over B, and, moreover, as it holds for A, which was assumed to be generic in C and independent from B, it defines an open subset V of C. (***)

Now choose a generic hyperplane H and let pr_{B}(C) be the projection of C from B to H. By the proof (below) that the degenerate case (†) cannot occur, and the argument above, we can find an open W ⊂ V, such that, for each x ∈ W, l_{x} is projected to l_{pr_{B}(x)}. We claim that;

For y ∈ pr_{B}(W), the l_{y} intersect in a point Q. (∗ ∗ ∗∗)

Let y_{1} and y_{2} be in pr_{B}(W), then we can find corresponding x_{1} and x_{2} in W such that < l_{By_{1}}, l_{y_{1}} >=< l_{Bx_{1}}, l_{x_{1}} > and < l_{By_{2}}, l_{y_{2}} >=< l_{Bx_{2}}, l_{x_{2}} >. As x_{1} and x_{2} lie in V, by (∗), we have that;

< l_{Bx_{1}}, l_{x_{1}} > ∩ < l_{Bx_{2}}, l_{x_{2}} >= l_{B}

Note that we can exclude the case that < l_{Bx}, l_{x} >=< l_{Bx}, l_{B} > for x ∈ V, as in this case C would clearly be contained in the plane < l_{Bx}, l_{B} >, use the fact that pr_{B}(W) would have the property that all its tangents intersected in a line l and the proof below that the degenerate case (†) cannot occur. Hence, it follows that;

l_{y_{1}} ∩ l_{y_{2}} = pr_{B}(l_{B}) = Q

as required. Now, to obtain the final contradiction, we need to show that (∗ ∗ ∗∗) cannot occur. This also covers the degenerate case (†). We show;

If C is a projective algebraic curve in P^{w}, with the property that there exists an open W ⊂ Nonsing(C) such that each l_{x} for x ∈ W intersects in a point Q, then C is a line l. (†)

Choose a generic hyperplane H and consider the projection pr_{Q}(C) of C from Q to H. If this projection is a point, then C is a line l. Hence, we may assume that pr_{Q}(C) is a projective algebraic curve in H. Suppose that x is generic in pr_{Q}(C), with pr_{Q}(y) = x and y generic in C. Then x and y are nonsingular and, moreover, if H_{λ} ⊂ H is any hyperplane passing through x, the corresponding hyperplane < H_{λ}, Q > passes through y. By the assumption, < H_{λ}, Q > contains the tangent line l_{y}. Hence, by the proof given above,

min{I_{italian}(y, C, < H_{λ}, Q >), I_{italian}(x, C, H_{λ})} ≥ 2.

Now, as x is nonsingular, we can find a hyperplane H_{λ} passing through x, not containing l_{x}. By the usual argument, given above, we obtain a contradiction. Therefore, (†) is shown.

As we assumed that our original C was not contained in a P^{2}, the proof of the lemma is shown. The lemma also holds in non-zero characteristic even though, surprisingly, the result (†) turns out to be false. We will deal with these problems in the final section.

□

Remarks 4.3. The proof of the above lemma is attributed to Castelnuovo in [10], but I have been unable to find a convenient reference.

Definition 4.4. We will define a correspondence Γ ⊂ C_{1} × C_{2}, where C_{1} and C_{2} are projective algebraic curves, to be generally biunivocal, if, for generic x ∈ C_{1} there exists a unique generic y ∈ C_{2} such that Γ(x, y) and Γ(x′, y) implies that x = x′ (*). We will say that Γ is biunivocal at x if (∗) holds.

We then have as an immediate consequence of the above that;

Lemma 4.5. Let C ⊂ P^{w}, (w ≥ 3), be a projective algebraic curve, not contained in any hyperplane section. Fix some hyperplane H. Then, if P is a generic point of the curve, the projection pr_{P} from P to H is generally biunivocal on C.

We now note the following;

Lemma 4.6. Let C ⊂ P^{w} be as in the above lemma. Fix a plane ω of dimension w − k ≥ 2. Then, if {P_{1}, …, P_{k}} are independent generic points of C, the projection from Ω =< P_{1}, …, P_{k} > to ω is generally biunivocal.

Proof. Choose a sequence of planes ω = ω_{0} ⊂ ω_{1} ⊂ … ⊂ ω_{k−1} such that dim(ω_{i}) = dim(ω) + i for i ≤ k − 1, with field of definition equal to that of ω. The projection pr_{P_{1}} from P_{1} to ω_{k−1} is generally biunivocal, and {pr_{P_{1}}(P_{2}), …, pr_{P_{1}}(P_{k})} define independent generic points on pr_{P_{1}}(C). Repeating the argument k times, we obtain that the composition pr_{P_{k}} ∘ … ∘ pr_{P_{1}} is biunivocal as a projection to ω. We claim that pr_{<P_{1},...,P_{k}>} = pr_{P_{k}}∘…∘pr_{P_{1}} (*). This can be checked for a generic point x of C. Suppose, inductively, that pr_{<P_{1},...,P_{i}>}(x) = pr_{P_{i}} ∘…∘pr_{P_{1}}(x) = z. Let u = pr_{P_{i+1}}(z), then there exists z′ =< P_{1}, …, P_{i+1} > ∩ω_{k−i} such that l_{zz′} ∩ω_{k−i−1} = u. We have that < P_{1}, …, P_{i}, x > ∩ω_{k−i} = z, hence < P_{1}, …, P_{i}, P_{i+1}, x > ∩ω_{k−i} = l_{zz′}, therefore < P_{1}, …, P_{i+1}, x > ∩ω_{k−i−1} = u. This shows that pr_{<P_{1},...,P_{i+1}>}(x) = u and, therefore, by induction, that pr_{<P_{1},...,P_{k}>}(x) = pr_{P_{k}} ∘…∘pr_{P_{1}}(x). Hence (∗) is shown. It follows immediately that the projection from Ω =< P_{1}, …, P_{k} > to ω is biunivocal on C as required. □

As a consequence, we note the following, which is of independent interest;

Lemma 4.7. Let C ⊂ P^{w} be as in the previous lemma. Let {P_{1}, …, P_{k}} be independent generic points on C for k ≤ w−1. Then the hyperplane < P_{1}, …, P_{k} > does not otherwise encounter C.

Proof. By the previous lemma, the projection pr_{<P_{1},...,P_{k−1}>} is generally biunivocal. Suppose there existed another intersection Q of < P_{1}, …, P_{k} > with C. Then pr_{<P_{1},...,P_{k−1}>}(Q) = pr_{<P_{1},...,P_{k−1}>}(P_{k}), contradicting the definition of generally biunivocal. □

Using Lemma 4.6, we obtain an alternative proof of Theorem 1.33;

Theorem 4.8. Let C ⊂ P^{w} be a projective algebraic curve, then C is birational to a plane projective curve.

Proof. By Lemma 4.6, we can find a generally biunivocal correspondence between C and C_{1} ⊂ P^{2}. In characteristic 0, we can therefore find U ⊂ C and V ⊂ C_{1} such that U ≅ V. This defines a birational map Φ_{Σ} : C ↭ C_{1}. □

We now use the techniques of this section to prove some further results which will be required later. We have first;

Lemma 4.9. Let C ⊂ P^{w} be a projective algebraic curve, not contained in any hyperplane section, and x ∈ C. Then there exists a plane Ω of dimension w − 3 and a plane ω of dimension 2 such that the projection pr from Ω to ω is generally biunivocal and, moreover, biunivocal at x.

Proof. Let H be a hyperplane containing x, then, by the assumption on C, it intersects C in a finite number r of points. It follows that we can find P generic in H such that l_{P x} does not otherwise encounter the curve C. Now choose a further hyperplane H′ not containing P. Let Q be generic on C with Q independent from P. Suppose that l_{P Q} intersects C in a new point R, (∗). Then {Q, R} must form a generic independent pair. Otherwise, as P ∈ l_{QR}, we would have that dim(P/Q) = 0 and therefore dim(P) = 0 which is a contradiction. Now, we can imitate the proof of Lemma 4.2 for the independent pair {Q, R} and the projection pr_{P} to obtain a contradiction. It follows that (∗) cannot occur, hence the projection pr_{P} is generally biunivocal and, moreover, by construction, biunivocal at x. Now, repeat this argument w − 2 times, to find a sequence of points {P_{1}, …, P_{w−2}} and planes {ω = ω_{1} ⊂ ω_{2} ⊂ … ⊂ ω_{w−1} = P^{w}} such that the projection pr_{P_{j}} from ω_{j+1} to ω_{j} is generally biunivocal on C and, moreover, biunivocal at x, for 1 ≤ j ≤ w − 2. It follows that the projection pr_{<P_{1},...,P_{w−2}>} from < P_{1}, …, P_{w−2} > to ω has the required property. □

Definition 4.10. Let C ⊂ P^{w} be a projective algebraic curve. Let pr be a projection from Ω to ω such that pr is biunivocal at x. We call < Ω, x > the axis of the projection at x.

We now refine Lemma 4.9 to ensure that the axis of projection is not in ”special position” with respect to x.

Definition 4.11. Special Position

Let C ⊂ P^{w} be a projective algebraic curve and x an s-fold point on C, (see Definition 3.1). Let Ω be a plane passing through x. We say that Ω is in special position if for every hyperplane H containing Ω;

I_{italian}(x, C, H) ≥ s + 1.

Lemma 4.12. Let hypotheses be as in Lemma 4.9. Let x be an s-fold point of C. Then we can obtain the conclusion of Lemma 4.9, with the extra requirement that the axis of projection < Ω, x > is not in special position.

Proof. As x is s-fold on C, we can find a hyperplane H passing through x such that I_{italian}(x, C, H) = s. Now imitate the proof of Lemma 4.9 by choosing a point P_{1}, generic on H such that l_{P_{1}x} does not otherwise encounter the curve C. Repeating the argument for the projected hyperplane pr_{P_{1}}(H) in P^{w−1}, we obtain a series {P_{1}, …, P_{w−2}} as in the conclusion of Lemma 4.9 and, by construction, the axis of projection < P_{1}, …, P_{w−2}, x >⊂ H. Hence, < Ω, x > is not in special position.

□

We can now prove;

Lemma 4.13. Let C ⊂ P^{w} be a projective algebraic curve. Suppose that x ∈ C and pr is a projection from Ω to ω = P^{2} satisfying the conclusion of Lemma 4.12. Then x is s-fold on C iff pr(x) is s-fold on pr(C).

Proof. Let V = {x ∈ C : pr is biunivocal at x} and W = pr(V). V and W are open subsets of C and pr(C) respectively and are in bijective correspondence. Let {H_{λ} : λ ∈ P ar_{H}} be the independent system Σ of lines in P^{2}. We then obtain a corresponding independent system Σ′ on P^{w} defined by {< Ω, H_{λ} >: λ ∈ P ar_{H}}. Clearly, Σ has no base points on pr(C). We claim that Σ′ has no base points on C. Suppose that there existed a base point y ∈ C for Σ′, then clearly pr(y) would be a base point for Σ on pr(C), which is a contradiction. We now claim that for y ∈ V and corresponding pr(y) ∈ W, that;

I_{italian}(y, C, < Ω, H_{λ} >) = I_{italian}(pr(y), pr(C), H_{λ}) (∗)

By the fact that Σ′ has no base points, and results of Section 2, we have that;

I_{italian}(y, C, < Ω, H_{λ} >) = I_{italian}^{Σ′}(y, C, < Ω, H_{λ} >)

Therefore, the result follows immediately from the definition of I_{italian}^{Σ′} and the fact that V and W are in bijective correspondence.

We now claim that deg(C) = deg(pr(C)). Suppose that deg(pr(C)) = d, then, by the proof of Lemma 2.4, we can find a generic plane H_{λ} intersecting pr(C) transversely at d distinct points inside W. By (∗), the corresponding hyperplane < Ω, H_{λ} > intersects C transversely at d distinct points inside V. Hence, by general properties of infinitesimals, deg(C) = d as well.

Now, let Σ′′ be the independent system defined by the lines in P^{2} passing through pr(x). By Bezout, it defines a g_{d}^{r}(Σ′′) on pr(C). Suppose that pr(x) is s_{1}-fold on pr(C), then, after removing the fixed point contribution at pr(x), we obtain a g_{d−s_{1}}^{r} on pr(C) without fixed points. Let Σ′′′ be the independent system defined by < Ω, Σ′′ >. It consists exactly of the hyperplanes passing through the axis of projection < Ω, x >. Again, by hyperspatial Bezout, it defines a g_{d}^{r}(Σ′′′) on C. As pr was assumed to be biunivocal at x, this g_{d}^{r}(Σ′′′) has a unique fixed point at x. Moreover, by the assumption on pr that the axis of projection is not in special position, if x is s_{2}-fold on C, then its fixed point contribution at x is s_{2}. Hence, after removing this contribution, we obtain a g_{d−s_{2}}^{r} on C without fixed points. Now, using Lemma 2.17, for generic λ, the weighted set W_{λ} for g_{d−s_{2}}^{r} consists of d − s_{2} points, each counted once, lying inside V. Using (∗), the corresponding V_{λ} for g_{d−s_{1}}^{r} consists of d − s_{2} points, each counted once, lying inside W. By Lemma 2.17 again, we must have that d − s_{1} = d − s_{2}, hence s_{1} = s_{2} as required. □

As an easy consequence of the above lemma, we have;

Lemma 4.14. Let C ⊂ P^{w} be a projective algebraic curve and suppose that x ∈ C, then x is non-singular iff x is not multiple.

Proof. Choose a projection pr as in Lemma 4.13. Then, x is not multiple iff pr(x) is not multiple. By Lemma 3.2, pr(x) is not multiple iff pr(x) is non-singular. In characteristic 0, using the fact that pr is generally biunivocal and an elementary model theoretic argument, we can find an inverse morphism ϕ : W′ → V′, where {W′, V′} are open subsets of {W, V } given in the previous lemma. It follows that V′ ≅ W′. As pr(x) is non singular, we can extend the morphism ϕ to include pr(x) in its domain, by biunivocity of pr at x, we must have that ϕ(pr(x)) = x. Hence, we may assume that x lies in V′ and pr(x) lies in W′. Therefore, pr(x) is non-singular iff x is non-singular. Hence, the result is shown.

□

Combining this result with Theorem 3.3, we then have the following;

Theorem 4.15. Let C ⊂ P^{w} be a projective algebraic curve, then C is birational to a non-singular projective algebraic curve C_{1} ⊂ P^{w′}.

We finish this section with the following application of the method of Conic Projections, the result will not be required later in the paper. Some part of the proof will require methods developed in Sections 5 and 6, we refer the reader to Definition 6.3 for the terminology ”node”. I have not seen a reasonable algebraic proof of this result.

Theorem 4.16. Let C ⊂ P^{w} be a projective algebraic curve, then C is birational to a plane projective algebraic with at most nodes as singularities.

Proof. We may, by the previous theorem, assume that C is non-singular. We first reduce to the case w = 3. That is, we claim that C is birational to a non-singular projective algebraic curve C′ ⊂ P^{3}. Let V_{3} be the variety of chords on C. That is;

V_{3} = \overline{\{⋃ l_{ab} : (a, b) ∈ C^{2} \ Δ\}}

We may assume that C is not contained in any plane of dimension 2 (†) and that w ≥ 4. We then claim that V_{3} has dimension 3. Let a ∈ C and define;

Cone_{a} = \overline{\{⋃ l_{ab} : b ∈ (C \ a)\}}

As C is irreducible, so is Cone_{a}. (use the fact that for any component W of Cone_{a}, {b ∈ (C \ a) : l_{ab} ⊂ W } is a closed subset of C \ a). Suppose that Cone_{a} has dimension 1. Then there exists b such that;

Cone_{a} = l_{ab}

It follows immediately that C is contained in l_{ab}, contradicting the hypothesis (†). Hence, Cone_{a} has dimension 2. Now, suppose that V_{3} has dimension 2. Then there exist finitely many points {a_{1}, …, a_{m}} such that; V_{3} = Cone_{a_{1}} ∪ … ∪ Cone_{a_{m}}

In particular, we can find distinct {a, b} ⊂ C such that Cone_{a} = Cone_{b}. Choose a′ ∈ C distinct from {a, b}. Then we may assume that l_{aa′} has infinite intersection with Cone_{b} \ δ(Cone_{b}) and therefore C ⊂ H_{aa′b}, contradicting the hypothesis (†). Hence, V_{3} has dimension 3 as required. Now define the tangent variety on C;

T ang(C) = {⋃ l_{a} : a ∈ C, l_{a} is the tangent line to C at a}

We claim that T ang(C) is a closed subvariety of V_{3} of dimension 2. In order to see that T ang(C) ⊂ V_{3}, it is sufficient to prove that l_{a} ⊂ Cone_{a}. Consider the following covers F, F^{∗} ⊂ (C \ {a}) × P^{w};

F (x, y) ≡ x ∈ (C \ {a}) ∧ y ∈ l_{ax}

F^{∗}(x, λ) ≡ x ∈ (C \ {a}) ∧ H_{λ} contains l_{ax}

Let F̄ and F̄^{∗} be the closures of these covers inside C ×P^{w}. We have the incidence relation I ⊂ P^{w} × P^{w^{∗}} given by;

I(x, λ) ≡ x ∈ H_{λ}

By definition, for x ∈ (C \ {a}), we have that;

I(F̄(x), F̄^{∗}(x));

Hence, this relation holds at {a} as well (∗). Now, by the proof of Theorem 6.7, the fibre F̄^{∗}(a) consists exactly of the hyperplanes H_{λ} containing l_{a}. Hence, by (∗), we must have that the fibre F̄(a) defines l_{a}. In order to complete the proof, observe that the projection pr : C × P^{w} → P^{w} defines a morphism from F̄ to Cone_{a}. Let;

Γ_{pr}(x, y, z) ⊂ F̄ × Cone_{a} ⊂ C × P^{w} × P^{w}

be the graph of this projection. Then, for x ∈ (C \ {a}), the fibre Γ_{pr}(x) = l_{ax} × l_{ax}. Hence, the fibre Γ_{pr}(a) = l_{a} × l_{a}. This proves that l_{a} ⊂ Cone_{a} as required. We clearly have that T ang(C) has dimension 2 if C is not contained in a line l, contradicting the hypothesis (†). The fact that T ang(C) is a closed subvariety of V_{3} follows immediately from the fact that T ang(C) is a closed subvariety of P^{w}.

Now choose a plane Ω of dimension w − 4 such that Ω ∩ V_{3} = ∅. Let pr be the projection from Ω to ω where ω is a plane of dimension 3 and Ω∩ω = ∅. Let pr(C) ⊂ ω be the projection of C. We claim that pr(C) is birational to C and non-singular. First, observe that pr is defined everywhere on C as C ⊂ V_{3} and Ω ∩ V_{3} = ∅. Secondly, we have that pr is biunivocal everywhere on C. For suppose that pr(x) = pr(x′), then < Ω, x >=< Ω, x′ >=< Ω, pr(x) >, hence l_{xx′} intersects Ω, contradicting the fact that Ω ∩ V_{3} = ∅. Finally, we have that for every x ∈ C, the axis of projection < Ω, x > is not in special position, see Definition 4.11. This follows from the fact that, as x is non-singular, < Ω, x > would only be in special position if it contained the tangent line l_{x}. As T ang(C) ⊂ V_{3}, this would contradict the fact that Ω∩V_{3} = ∅. Now, by Lemmas 4.13 and 4.14, we must have that pr(x) is non-singular for every x ∈ C, that is pr(C) is non-singular. We may, therefore, using the argument of Lemma 4.14 invert the morphism pr to obtain an isomorphism between C and pr(C). In particular, C and pr(C) are birational.

We now consider the curve C′ = pr(C) ⊂ P^{3}. In order to prove the theorem, it will be sufficient to show that C′ is birational to a plane projective curve with at most nodes as singularities. We now define the following 5 varieties, we use the notation l to denote a line and P to define a 2-dimensional plane;

(i). T angent(C′) = {⋃ l_{a} : a ∈ C, l_{a} the tangent line at C}

(ii). T risecant(C′) = \overline{\{⋃ l_{abc} : (a, b, c) ∈ C^{3} distinct, l_{abc} ⊃ \{a, b, c\}\}}

By a bitangent plane P_{ab}, we mean a hyperplane passing through the tangent lines l_{a} and l_{b} for distinct {a, b} in C. By an osculatory plane P_{a}, we refer the reader to Definition 6.3.

We define Supp(P_{ab}) = {a, b} and Supp(P_{a}) = C′ ∩ P_{a}. We then consider;

(iii). Bitangent Chord(C′) = \overline{\{⋃ l_{ab} : (a, b) ∈ (C^{2} \ Δ), l_{ab} ⊃ Supp(P_{ab})\}}

(iv). Osculatory Chord(C′) = \overline{\{⋃_{a∈C} l_{ab} : (a, b) ∈ (Supp(P_{a})^{2} \ Δ)\}}.

By Remark 6.6, there exists a finite set W ⊂ C′, consisting of points which are the origins of non-ordinary branches. We let;

(v). Singular Cone(C′) = ⋃_{y∈W} Cone_{y}

As we have already seen, T angent(C′) defines a 2-dimensional algebraic variety, unless C′ is contained in a line, which we may assume is not the case. By Lemma 4.2, there are no trisecant lines passing through an independent generic pair (a, b) of C^{′2}, unless C′ is contained in a plane, which again we may exclude. The statement D(x, y) ⊂ C^{′2} \ Δ, given by;

D(x, y) ≡ {(x, y) ∈ C^{′2} : (x ≠ y)∧ ∄w(w ≠ x∧w ≠ y∧w ∈ C′∩l_{xy})}

is clearly constructible and has the property that \overline{D} = C^{′2}. This clearly implies that dim(T risecant(C′)) ≤ 2. We now claim that, if (a, b) is an independent generic pair in C^{′2}, there is no bitangent plane P_{a,b}. Let Σ = {H_{λ} : λ ∈ P^{3∗}} be the system of hyperplanes in P^{3}. Σ has no fixed points on C′, hence the linear condition that H_{λ} passes through l_{a} has codimension 2 and defines a 1-dimension family Σ_{1} ⊂ Σ. As we may assume that C′ is not contained in any hyperplane section,Σ_{1} has finite intersection with C′. As b is independent generic from a, it cannot be a base point for the new system Σ_{1}. Hence, using the results of Theorems 6.2 and 6.5, the condition that H_{λ} ∈ Σ_{1} passes through l_{b} is also a codimension 2 linear condition. As Σ_{1} is 1-dimensional, this implies the claim. Now consider the statement D′(x, y) ⊂ C^{′2} \ Δ, given by;

D′(x, y) ≡ {(x, y) ∈ C^{′2} : (x ≠ y)∧ ∄λ(H_{λ} ⊃ l_{a} ∪ l_{b})}

Again, this is constructible and has the property that \overline{D′} = C^{′2}. This clearly implies that dim(Bitangent Chord(C′)) ≤ 2. For any given a ∈ C, there exists a unique osculatory plane P_{a} passing through a. Hence, there exist finitely many lines of the form l_{ab} for b ∈ (C^{2}\Δ)∩Supp(P_{a}). This clearly implies that dim(Osculatory Chord(C′)) ≤ 2. Finally, by the above consideration, we have that dim(Singular Cone(C′)) = 2

We now choose a point P in P^{3} such that P lies outside the above defined varieties. This is clearly possible as they all have dimension at most 2. Let ω be a plane of dimension 2 not containing P and let pr_{P} (C′) be the projection of C′ from P to ω. We claim that pr_{P} (C′) is birational to C′ with at most nodes as singularities. First, suppose that y ∈ pr_{P} (C′) is singular, then, by Lemma 4.14, it must be multiple. By Lemma 4.13 and the fact that C′ is non-singular, this can only occur if either |pr_{P}^{−1}(y)| ≥ 2 or there exists a unique x such that pr_{P} (x) = y and the axis of projection < P, x > is in special position. We may exclude the second possibility on the grounds that P is disjoint from T ang(C′). We may therefore assume that |pr_{P}^{−1}(y)| ≥ 2. Now, we may exclude the possibility that |pr_{P}^{−1}(y)| ≥ 3, on the grounds that P is disjoint from T risecant(C′). Hence, we may assume that |pr_{P}^{−1}(y)| = 2. Now, as C′ is non-singular, by Definition 5.2, there exist exactly 2 branches centred at y on pr_{P} (C′). Moreover, we may assume that both elements of pr_{P}^{−1}(y) = {x_{1}, x_{2}} are the origins of ordinary branches, otherwise P would lie inside Singular Cone(C′). If P is situated on the osculatory plane P_{x_{1}} of say x_{1}, then (x_{1}x_{2}) ⊂ Supp(P_{x_{1}}) and, hence, as P ∈ l_{x_{1}x_{2}}, we would have that P lies in Osculatory Chord(C′), which is not the case. Hence, we may apply Theorem 6.4, to obtain that both branches have character (1, 1). Finally, let l_{a} and l_{b} be the tangent lines to the 2 branches {γ_{y}^{1}, γ_{y}^{2}} at y. We claim that l_{a} and l_{b} are distinct (∗). By definition, we have that;

I_{italian}(y, γ_{y}^{1}, pr_{P} (C′), l_{a}) = 2

Hence, using Definition 5.15 and Lemma 5.17, we can find an infinitesimal variation l′_{a} of l_{a} intersecting γ_{y}^{1} in distinct points {y′, y′′}. By Definition 5.15, we can find distinct points {x′_{1}, x′′_{1}} in C′∩V_{x_{1}} such that pr_{P} (x′_{1}) = y′ and pr_{P} (x′′_{1}) = y′′. We clearly have that that < l′_{a}, P > is an infinitesimal variation of < l_{a}, P > which intersects C′ in {x′_{1}, x′′_{1}}. Therefore;

I_{italian}(x_{1}, C′, < l_{a}, P >) ≥ 2

and hence, by previous arguments, < l_{a}, P > contains the tangent line l_{x_{1}}. As the axis of projection l_{P x_{1}} was not in special position, we must have that pr_{P} (l_{x_{1}}) = l_{a}. Similarly, pr_{P} (l_{x_{2}}) = l_{b}. Now, if l_{a} = l_{b}, we would have that < l_{a}, P > is a bitangent plane to C′, l_{x_{1}x_{2}} is a bitangent chord and P would belong to the variety Bitangent Chord(C′). As this is not the case, the claim (∗) is proved. This completes the theorem.

□

5. A Theory of Branches for Algebraic Curves

We now develop a theory of branches for algebraic curves, using the techniques of the previous sections.

Let C ⊂ P^{w} be a projective algebraic curve. Then, by Theorem 4.15, we can find C^{ns} ⊂ P^{w_{1}} which is non-singular and birational to C. Let Φ^{ns} : C^{ns} ↭ C be the birational map between C^{ns} and C. As C^{ns} is non-singular, Φ^{ns} extends to a totally defined morphism from C^{ns} to C. As usual, we let Γ_{Φ^{ns}} denote the graph of the correspondence between C^{ns} and C. It has the property that, given any x ∈ C^{ns}, there exists a unique y ∈ C such that Γ_{Φ^{ns}}(x, y). We claim the following;

Lemma 5.1. Let C_{1} ⊂ P^{w_{1}} and C_{2} ⊂ P^{w_{2}} be any two non-singular birational models of C, with corresponding morphisms Φ_{1}^{ns} and Φ_{2}^{ns}. Then there exists a unique isomorphism Φ : C_{1} ↔ C_{2} with the property that Φ_{2}^{ns} ∘ Φ = Φ_{1}^{ns} and Φ_{1}^{ns} ∘ Φ^{−1} = Φ_{2}^{ns}.

Proof. As Φ_{2}^{ns} is a birational map, we can invert it to give a birational map Φ_{2}^{ns}^{−1} : C ↭ C_{2}. Then the composition Φ_{2}^{ns}^{−1} ∘ Φ_{1}^{ns} : C_{1} ↭ C_{2} is birational as well. As C_{1} is non-singular, we can extend this birational map to a totally defined morphism Φ : C_{1} → C_{2}. By the same argument, we can find a totally defined morphism Ψ : C_{2} → C_{1} with the property that Ψ inverts Φ as a birational map. We claim that Ψ inverts Φ as a morphism. We have that Ψ ∘ Φ : C_{1} → C_{1} is a morphism with the property that there exists an open U ⊂ C_{1} such that Ψ ∘ Φ|U = Id_{U}. Then Graph(Ψ ∘ Φ) ⊂ C_{1} × C_{1} is closed irreducible and intersects the diagonal Δ in an open dense subset. Therefore, Graph(Ψ ∘ Φ) = Δ, hence Ψ ∘ Φ = Id_{C_{1}}. Similarly, one shows that Φ ∘ Ψ = Id_{C_{2}}. Therefore, Φ is an isomorphism. By construction, Φ_{1}^{ns} and Φ_{2}^{ns} ∘ Φ agree as birational maps and are totally defined, hence, by a similar argument, they agree as morphisms. Similarly, Φ_{2}^{ns} and Φ_{1}^{ns} ∘ Φ^{−1} agree as morphisms. For the uniqueness statement, use the fact that any 2 isomorphisms Φ_{1} and Φ_{2}, satisfying the properties of Φ, would agree on an open subset U of C_{1}, hence must be identical.

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Now fix a point O of C. Let Γ_{Φ^{ns}} ⊂ C^{ns} × C be the graph of the correspondence defined above. We denote by {O_{1}, …, O_{t}} the fibre Γ_{Φ^{ns}}(x, O). Note that, by the previous lemma, if we are given 2 nonsingular models with correspondences Γ_{Φ_{1}^{ns}} and Γ_{Φ_{2}^{ns}}, then we have a correspondence O_{1} ∼ O′_{1}, O_{2} ∼ O′_{2}, …, O_{t} ∼ O′_{t} between the fibres Γ_{Φ_{1}^{ns}}(x, O) and Γ_{Φ_{2}^{ns}}(x, O), given by O_{j} ∼ O′_{j} iff Φ(O_{j}) = O′_{j}, where Φ is the isomorphism given by the previous lemma. Moreover, as Φ is an isomorphism, we also have a correspondence of infinitesimal neighborhoods V_{O_{1}} ∼ V_{O′_{1}}, …, V_{O_{t}} ∼ V_{O′_{t}}, given by V_{O_{j}} ∼ V_{O′_{j}} iff Φ : V_{O_{j}} ≅ V_{O′_{j}}, here we mean that Φ is a bijection of sets. We now make the following definition; Definition 5.2. Let C ⊂ P^{w} be a projective algebraic curve. Suppose that O ∈ C. For 1 ≤ j ≤ t, we define the branches γ_{O}^{j} at O to be the equivalence classes [V_{O_{j}}] of the infinitesimal neighborhoods of O_{j} in the fibre Γ_{Φ^{ns}}(x, O) of any non-singular model C^{ns} of C. We define γ_{O} to be the union of the branches at O.

Remarks 5.3. Note that the definition does not depend on the choice of a non-singular model C^{ns}, however, for computational purposes, it is convenient to think of a branch γ as the infinitesimal neighborhood of some O_{j} in a fixed non-singular model C^{ns}.

We now have the following lemmas concerning branches;

Lemma 5.4. Let C ⊂ P^{w} and O ∈ C be an s-fold point. Then O is the origin of at most s-branches. In particular, a non-singular point is the origin of a single branch.

Proof. Suppose that O is the origin of t branches. Then there exists a non-singular model C^{ns} and a birational map Φ_{Σ} : C^{ns} → C such that Φ_{Σ}^{−1}(O) = {O_{1}, …, O_{t}}. By a slight extension to Remarks 1.32, and the fact that {O_{1}, …, O_{t}} are non-singular, we may assume that Base(Σ) is disjoint from this set (*). Let Σ define the system of hyperplanes in P^{w}. It defines a g_{d}^{r}(Σ) without fixed points on C, where d = deg(C). By Lemma 2.32, there is a corresponding g_{d+I}^{r}(Σ) on C^{ns}, where I is the total fixed point contribution at Base(Σ). Now let Σ′ ⊂ Σ define the system of hyperplanes passing through O. It defines a g_{d}^{r−1}(Σ′) on C with corresponding g_{d+I}^{r−1}(Σ′) on C^{ns}. As O is s-fold on C, we can find g_{d−s}^{r−1} ⊂ g_{d}^{r−1}(Σ′) on C without fixed points. By Lemma 2.31, we can find a corresponding g_{d−s}^{r−1} ⊂ g_{d+I}^{r−1}(Σ′) on C^{ns}, without fixed points (**). Now each ϕ_{λ} in Σ′ must pass through {O_{1}, …, O_{t}}. Moreover, the fixed point contribution from Σ′ at Base(Σ) must be at least I. Hence, by the assumption (*), the total fixed point contribution from Base(Σ′) must be at least t+I. Hence, by (∗∗), d−s ≤ (d+I)−(t+I), therefore t ≤ s. The lemma is proved.

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We now make the following definition;

Definition 5.5. Let C_{1} ⊂ P^{w_{1}}, C_{2} ⊂ P^{w_{2}}, C_{3} ⊂ P^{w_{3}} be birational projective algebraic curve with correspondences Γ_{Φ_{1}} ⊂ C_{1} × C_{2} and Γ_{Φ_{2}} ⊂ C_{2} × C_{3}. We define the composition Γ_{Φ_{2}} ∘ Γ_{Φ_{1}} ⊂ C_{1} × C_{3} to be;

Γ_{Φ_{2}} ∘ Γ_{Φ_{1}}(xz) ≡ ∃y(y ∈ C_{2} ∧ Γ_{Φ_{1}}(xy) ∧ Γ_{Φ_{2}}(yz))

Lemma 5.6. Let hypotheses be as in the previous lemma, then if Φ_{1} : C_{1} → C_{2} and Φ_{2} : C_{2} → C_{3} are birational maps;

Γ_{Φ_{2}} ∘ Γ_{Φ_{1}} = Γ_{Φ_{2}∘Φ_{1}}

Proof. The proof is a straightforward consequence of the fact that both correspondences obviously agree on a Zariski open subset. □

Lemma 5.7. Birational Invariance of Branches

Let C_{1} ⊂ P^{w_{1}} and C_{2} ⊂ P^{w_{2}} be birational projective algebraic curves with correspondence Γ_{[Φ]} ⊂ C_{1} ×C_{2}. Fix O ∈ C_{2} and let {P_{1}, …, P_{s}} = Γ_{[Φ]}(x, O). Then, [Φ] induces an injective map;

[Φ]^{∗} : γ_{O} → ⋃_{1≤k≤s} γ_{P_{k}}

and, moreover;

[Φ]^{∗} : ⋃_{O∈C_{2}} γ_{O} → ⋃_{O∈C_{1}} γ_{O}

is a bijection, with inverse given by [Φ^{−1}]^{∗}.

Proof. We first claim that there exists a non-singular model C^{ns} of C_{1} and C_{2} with morphisms Φ_{1} : C^{ns} → C_{1} and Φ_{2} : C^{ns} → C_{2} such that Φ_{2} = Φ ∘ Φ_{1} and Φ_{1} = Φ^{−1} ∘ Φ_{2} as birational maps (*). Choose a non-singular model C^{ns} of C_{1} with morphism Φ_{1} : C^{ns} → C_{1}. Then Φ∘Φ_{1} defines a birational map from C^{ns} to C_{2}. As C^{ns} is non-singular, it extends(uniquely) to a birational morphism Φ_{2} : C^{ns} → C_{2}. Clearly, {C^{ns}, Φ_{1}, Φ_{2}} have the property (∗). In order to define the map [Φ]_{∗}, let O ∈ C_{2} and suppose that γ_{O}^{j} is a branch corresponding to the equivalence class [V_{O_{j}}], where O_{j} lies in the fibre Φ_{2}^{−1}(O). We claim that Φ_{1}(O_{j}) lies in the fibre Γ_{[Φ]}(x, O) (∗∗). As Φ∘Φ_{1} = Φ_{2} as birational maps, it follows, by Lemma 1.21, that Γ_{Φ∘Φ_{1}} = Γ_{Φ_{2}}. However, by the previous lemma, Γ_{Φ∘Φ_{1}} = Γ_{Φ} ∘ Γ_{Φ_{1}}, which gives (∗∗). Then, if Φ_{1}(O_{j}) = P_{k}, and Φ_{1}^{−1}(P_{k}) = {P′_{k1}, …, P′_{kr}, …, P′_{kl}}, for 1 ≤ r ≤ l, we must have that [V_{O_{j}}] = [V_{P′_{kr}}], for some r. Hence, we can set [Φ]^{∗}(γ_{O}^{j}) = γ_{P_{k}}^{r}. One needs to check that this definition does not depend either on the choice of the non-singular model C^{ns} or the choice of the birational map Φ representing the correspondence Γ_{[Φ]}. For the first claim, after fixing a choice of Φ, note that by the requirement (∗), Φ_{1} is uniquely determined by the choice of {C^{ns}, Φ_{2}}. By Lemma 5.1, for any 2 choices {C_{1}^{ns}, Φ_{2}^{1}} and {C_{2}^{ns}, Φ_{2}^{2}}, there exists a connecting isomorphism Θ : (C_{1}^{ns}, Φ_{2}^{1}) ↔ (C_{2}^{ns}, Φ_{2}^{2}), which must therefore be a connecting isomorphism Θ : (C_{1}^{ns}, Φ_{1}^{1}) ↔ (C_{2}^{ns}, Φ_{1}^{2}). The result then follows immediately by definition of the branch as an equivalence class of infinitesimal neighborhoods. For the second claim, after fixing a choice of {C^{ns}, Φ_{1}, Φ_{2}} and replacing Φ by an equivalent Φ′, one still obtains (∗) for the original {C^{ns}, Φ_{1}, Φ_{2}} and the result follows trivially. The rest of the lemma now follows by proving that [Φ^{−1}]^{∗} inverts [Φ]^{∗}. This is trivial to check using the particular choice {C^{ns}, Φ_{1}, Φ_{2}, Φ}. □

Remarks 5.8. Note that points are not necessarily preserved by birational maps, but, by the above, branches are always preserved.

We now refine the Italian definition of intersection multiplicity, in order to take into account the above construction of a branch. Suppose that C ⊂ P^{w} is a projective algebraic curve. We let P ar_{F} be the projective parameter space for all hypersurfaces of a given degree e. Now, fix a particular form F_{λ} of degree e, such that F_{λ} has finite intersection with C. The condition that a hypersurface of degree e contains C is clearly linear on P ar_{F}, (use Theorem 2.3 to show that the condition is equivalent to the plane condition P on P ar_{F} of passing through de + 1 distinct points on the curve C.) Now fix a maximal linear system Σ containing F_{λ}, such that Σ has empty intersection with P. It is trivial to check that the corresponding g_{n}^{r}(Σ) on C has no fixed points. Now fix a non-singular model C^{ns} of C, with corresponding birational morphism Φ_{Σ′}. By the transfer result Lemma 2.31, we obtain a corresponding g_{n}^{r} without fixed points on C^{ns}. Let Card(O, V_{λ}, g_{n}^{r}) denote the number of times O ∈ C^{ns} is counted for this g_{n}^{r} in the weighted set V_{λ}. We then define;

Definition 5.9. I_{italian}(p, γ_{p}^{j}, C, F_{λ}) = Card(p_{j}, V_{λ}, g_{n}^{r}) (∗)

where the branch γ_{p}^{j} corresponds to [V_{p_{j}}] in the fibre Γ_{[Φ_{Σ′}]}(x, p).

Remarks 5.10. One may, without loss of generality, assume that Base(Σ′) is disjoint from the fibre Γ_{[Φ_{Σ′}]}(x, p). In which case, the definition becomes the more familiar;

I_{italian}(p, γ_{p}^{j}, C, F_{λ}) = Card(C^{ns}∩\overline{F}_{λ′}∩V_{p_{j}}), λ′ ∈ V_{λ} generic in P ar_{Σ}.

Here, we have used the notation \{\overline{F}_{λ} : λ ∈ P ar_{Σ}\} to denote the family of ”lifted” forms on C^{ns} corresponding to the family {F_{λ} : λ ∈ P ar_{Σ}} on C, by way of the birational morphism Φ_{Σ′}.

Lemma 5.11. Given Σ containing F_{λ}. The definition (∗) does not depend on the choice of the non-singular model C^{ns} and birational morphism Φ_{Σ′}.

Proof. We divide the proof into the following 2 cases;

Case 1. (C^{ns}, Φ) is fixed and we have 2 presentations Φ_{Σ_{1}} and Φ_{Σ_{2}} of the birational morphism Φ, as given by Lemma 1.20, such that Base(Σ_{1}) possibly includes some of the points in Γ_{[Φ]}(x, p), while Base(Σ_{2}) is disjoint from this set.

Let V = V_{Φ_{Σ_{1}}} ∩ V_{Φ_{Σ_{2}}} with corresponding W ⊂ C. Denote the weighted sets of the 2 given g_{n}^{r} on C^{ns},in the definition (∗), corresponding to the presentations Φ_{Σ_{1}} and Φ_{Σ_{2}}, by {V_{λ}^{1}} and {V_{λ}^{2}}. By the proof of Lemma 2.31, if x ∈ V, then x is counted s-times in V_{λ}^{1} iff x is counted s-times in V_{λ}^{2} (†). Now, we claim that Card(p_{j}, V_{λ}^{1}, g_{n}^{r}) = Card(p_{j}, V_{λ}^{2}, g_{n}^{r}). This follows by (†) and an easy application of, say Lemma 2.16, to witness both these cardinalities in the canonical set V.

Case 2. We have 2 non-singular models (C_{1}^{ns}, Φ_{1}) and (C_{2}^{ns}, Φ_{2}), with presentations Φ_{Σ_{1}} and Φ_{Σ_{2}}, such that Base(Σ_{1}) is disjoint from Γ_{[Φ_{1}]}(x, p) and Base(Σ_{2}) is disjoint from Γ_{[Φ_{2}]}(x, p).

Using Lemma 5.1, we can find an isomorphism Φ : C_{1} → C_{2}, such that Φ_{2} ∘ Φ = Φ_{1}. Let {G_{λ} : λ ∈ P ar_{Σ}} denote the lifted forms on C_{1}^{ns} corresponding to the morphism Φ_{Σ_{1}} and {H_{λ} : λ ∈ P ar_{Σ}} denote the lifted forms on C_{2}^{ns} corresponding to the morphism Φ_{Σ_{2}}. We need to check that;

Card(C_{1}^{ns}∩G_{λ′}∩V_{p_{j}}) = Card(C_{2}^{ns}∩H_{λ′}∩V_{q_{j}}), λ′ ∈ V_{λ}∩P ar_{Σ} generic.

where Φ(p_{j}) = q_{j} and {p_{j}, q_{j}} in {C_{1}^{ns}, C_{2}^{ns}} respectively, correspond to the branch γ_{p}^{j}.

Suppose that Card(C_{2}^{ns} ∩ H_{λ′} ∩ V_{q_{j}}) = n. We may, without loss of generality, assume that Base(Φ_{Σ_{3}}) is disjoint from Γ_{[Φ_{1}]}(x, p) in a particular presentation Φ_{Σ_{3}} of the morphism Φ (†). Hence, Base(Φ_{Σ_{2}} ∘ Φ_{Σ_{3}}) is disjoint from Γ_{[Φ_{1}]}(x, p) as well (††). Let {Φ_{Σ_{3}}^{∗}H_{λ} : λ ∈ P ar_{Σ}} denote the lifted forms on C_{1}^{ns} corresponding to this presentation. By the fact that Φ is an isomorphism and (†), we have Card(C_{1}^{ns} ∩Φ^{∗}H_{λ′} ∩ V_{p_{j}}) = n. Now let V = V_{Φ_{Σ_{1}}} ∩ V_{Φ_{Σ_{2}}∘Φ_{Σ_{3}}}. By (††) and Lemma 2.13, we may witness Card(C_{1}^{ns} ∩ Φ^{∗}H_{λ′} ∩ V_{p_{j}}) = n inside the canonical set V. Now, using the fact that Φ_{2} ∘ Φ = Φ_{1} as birational maps, and Lemma 2.31, it follows that Card(C_{1}^{ns} ∩ G_{λ′} ∩ V_{p_{j}}) = n.

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Lemma 5.12. The definition (∗) does not depend on the choice of a maximal independent system Σ containing F_{λ}, having finite intersection with C.

Proof. Let Σ be a maximal independent system containing F_{λ}. We claim first that Σ has no base points on C, (†). For suppose that w ∈ Base(Σ) ∩ C. Let F_{μ} be any form of degree e having finite intersection with C. Then < F_{μ}, Σ > defines a system of higher dimension, hence must intersect H in a point. That is, we can find parameters {α, β} and a fixed F_{λ} in Σ such that αF_{μ} + βF_{λ} contains C. It follows immediately that w must also be a base point for F_{μ}. Clearly, we can find a form of degree e, having finite intersection with C, which doesn’t contain w. This gives a contradiction and proves (†). Now, given a choice of Σ containing F_{λ}, by the proof of the previous lemma, we may assume that;

I_{italian}(p, γ_{p}^{j}, C, F_{λ}) = Card(C^{ns} ∩\overline{F_{λ′}} ∩V_{p_{j}}), λ′ ∈ V_{λ} generic in P ar_{Σ}.

As p is not a base point for Σ on C, it follows immediately that p_{j} is not a base point for Σ on C^{ns}. Hence, by Lemma 2.12, we have that;

I_{italian}(p, γ_{p}^{j}, C, F_{λ}) = I_{italian}(p_{j}, C^{ns}, \overline{F_{λ}})

Clearly, this equality does not depend on the particular choice of Σ containing F_{λ} but only on the presentation of the morphism Φ : C^{ns} → C. The result follows.

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Following from the definition of the Italian intersection multiplicity at a branch, we obtain a more refined version of Bezout’s theorem;

Theorem 5.13. Hyperspatial Bezout, Branched Version

Let C ⊂ P^{w} be a projective algebraic curve of degree d and F_{λ} a hypersurface of degree e, intersecting C in finitely many points. Let p be a point of intersection with branches given by {γ_{p}^{1}, …, γ_{p}^{n}}. Then;

I_{italian}(p, C, F_{λ}) = ∑_{1≤j≤n} I_{italian}(p, γ_{p}^{j}, C, F_{λ})

In particular;

∑_{p∈C∩F_{λ}} ∑_{1≤j≤n_{p}} I_{italian}(p, γ_{p}^{j}, C, F_{λ}) = de

Proof. Fix a non-singular model (C^{ns}, Φ) of C. Let Φ_{Σ′} be a particular presentation of the morphism Φ such that Base(Φ_{Σ′}) is disjoint from the fibre Γ_{Φ}(x, p) = {p_{1}, …, p_{n}} (∗). Let Σ be a linear system containing F_{λ} such that Σ has finite intersection with C and;

I_{italian}(p, C, F_{λ}) = I_{italian}^{Σ}(p, C, F_{λ})

Let V_{Φ_{Σ′}} ⊂ C^{ns} and W_{Φ_{Σ′}} ⊂ C be the canonical sets associated to the morphism Φ_{Σ′}. By Lemma 2.13, we can witness m = I_{italian}(p, C, F_{λ}) by transverse intersections {x_{1}, …, x_{m}} = C∩F_{λ′}∩V_{p} inside the canonical set W_{Φ_{Σ′}}, for λ′ generic in P ar_{Σ}. Again, by Lemma 2.13 and (∗), we can find λ′ such that for each p_{j} ∈ Γ_{[Φ]}(x, p), m_{j} = I_{italian}(p, γ_{p}^{j}, C, F_{λ}) is witnessed by transverse intersections \{y_{j}^{1}, …, y_{j}^{m_{j}}\} = C^{ns} ∩ \overline{F}_{λ′} ∩ V_{p_{j}} inside the canonical set V_{Φ_{Σ′}}. We need to show that m_{1}+…+m_{j} +…+ m_{n} = m. By properties of infinitesimals and the fact that {p_{1}, …, p_{n}} are distinct, the sets {y_{1}^{1}, …, y_{1}^{m_{1}}}, …, {y_{n}^{1}, …, y_{n}^{m_{n}}} are disjoint. If y_{i}^{j} belongs to one of these sets, then y_{i}^{j} ∈ V_{p_{i}}, hence Φ_{Σ′}(y_{i}^{j}) ∈ V_{p}. Moreover, Φ_{Σ′}(y_{i}^{j}) ∈ C ∩ F_{λ′}. It follows that y_{i}^{j} corresponds to a unique x_{k} in {x_{1}, …, x_{m}}. As each y_{i}^{j} lies in V_{Φ_{Σ′}}, this clearly gives an injection from {y_{1}^{1}, …, y_{1}^{m_{1}}, …, y_{n}^{1}, …, y_{n}^{m_{n}}} to {x_{1}, …, x_{m}}. Hence, m_{1} + … + m_{n} ≤ m. To prove equality, suppose that x_{k} lies in {x_{1}, …, x_{m}}. Then there exists a unique y_{k} with Φ_{Σ′}(y_{k}) = x_{k}. By a similar argument to the above, y_{k} must appear in one of the sets {y_{1}^{1}, …, y_{1}^{m_{1}}}, …, {y_{n}^{1}, …, y_{n}^{m_{n}}}. This gives the result.

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Remarks 5.14. One may use this version of Bezout’s theorem in order to develop a more refined theory of g_{n}^{r}.

We also simplify the branch terminology for later applications; Definition 5.15. Let C ⊂ P^{w} be a projective algebraic curve and C^{ns} ⊂ P^{w_{1}} some non-singular birational model with birational morphism Φ^{ns} : C^{ns} → C. Let γ_{p}^{j} correspond to the infinitesimal neighborhood V_{p_{j}} of p_{j} in Γ_{Φ^{ns}}(x, p). Then we will also define the branch γ_{p}^{j} by the formula;

γ_{p}^{j} := {x ∈ C : ∃y(Γ_{Φ^{ns}}(y, x) ∧ y ∈ V_{p_{j}})}

Remarks 5.16. Note that the definition uses the language L_{spec}, and, in particular, is not algebraic.

We have the following;

Lemma 5.17. Definition 5.15 does not depend on the choice of nonsingular model C^{ns} and morphism Φ^{ns}. Lemma 5.7 may be reformulated replacing the old definition of a branch with Definition 5.15. Finally, we have, with hypotheses as for the old definition of intersection multiplicity at a branch, that;

I_{italian}(p, γ_{p}^{j}, C, F_{λ}) = Card(C∩F_{λ′}∩γ_{p}^{j}), λ′ ∈ V_{λ} generic in P ar_{Σ} (∗)

Proof. The first part follows immediately from Lemma 5.1 and the fact that all the data of the lemma may be taken inside a standard model. The second part is similar, follow through the proof of Lemma 5.7. The final part may be checked by following carefully through the proofs of Lemmas 5.11 and 5.12. □

Remarks 5.18. Note that we could not have simplified the above presentation by taking (∗) as our original definition of intersection multiplicity at a branch. The main reason being that the arguments on Zariski multiplicities require us to count intersections inside C ∩ V_{p}, rather than the smaller L_{spec} definable C ∩ γ_{p}^{j}.

We now reformulate the preliminary definitions of Section 2 in terms of branches.

Let C ⊂ P^{w} be a projective algebraic curve and let Σ be a linear system, having finite intersection with C. Let g_{n′}^{r}(Σ) be defined by this linear system Σ and let g_{n}^{r} ⊂ g_{n′}^{r} be obtained by removing its fixed point contribution. Fix a non-singular model (C^{ns}, Φ) of C, with corresponding presentation Φ_{Σ′}. By the transfer result Lemma 2.31, we obtain a corresponding g_{n}^{r} without fixed points on C^{ns}. Let Card(O, V_{λ}, g_{n}^{r}) denote the number of times O ∈ C^{ns} is counted for this g_{n}^{r} in the weighted set V_{λ}. We then make the following definition;

Definition 5.19. I_{italian}^{Σ,mobile}(p, γ_{p}^{j}, C, ϕ_{λ}) = Card(p_{j}, V_{λ}, g_{n}^{r}) (∗)

where the branch γ_{p}^{j} corresponds to [V_{p_{j}}] in the fibre Γ_{[Φ_{Σ′}]}(x, p).

As before, one needs the following lemma;

Lemma 5.20. The definition (∗) does not depend on the choice of non-singular model C^{ns} and birational morphism Φ_{Σ′}.

Proof. The proof is the same as Lemma 5.11, we leave the details to the reader. □

We now make the following definition;

Definition 5.21. Let hypotheses be as in Definition 5.19, then we define;

I_{italian}^{Σ}(p, γ_{p}^{j}, C, ϕ_{λ}) = I_{italian}^{Σ,mobile}(p, γ_{p}^{j}, C, ϕ_{λ}) + 1 if p ∈ Base(Σ)

I_{italian}^{Σ}(p, γ_{p}^{j}, C, ϕ_{λ}) = I_{italian}^{Σ,mobile}(p, γ_{p}^{j}, C, ϕ_{λ}) if p ∉ Base(Σ)

Lemma 5.22. Let notation be as in the previous definition. As in Remarks 5.10, if (C^{ns}, Φ) is a non-singular model of C, with presentation Φ_{Σ′} such that Base(Σ′) is disjoint from the fibre Γ_{[Φ_{Σ′}]}(x, p), then we have that;

I_{italian}^{Σ}(p, γ_{p}^{j}, C, ϕ_{λ}) = Card(C^{ns}∩\overline{ϕ_{λ′}} ∩V_{p_{j}}), λ′ ∈ V_{λ} generic in P ar_{Σ}.

where again we have used the notation \{\overline{ϕ_{λ}} : λ ∈ P ar_{Σ}\} to denote the family of ”lifted” forms, as in Remarks 5.10.

Proof. We divide the proof into the following cases;

Case 1. p ∉ Base(Σ).

Then, by Definition 5.21, we have that;

A THEORY OF BRANCHES FOR ALGEBRAIC CURVES 67

I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ}) = I_{italian}^{Σ,mobile}(p, C, γ_{p}^{j}, ϕ_{λ}) (1). By the assumption on Σ′, we have that p_{j} ∉ Base(Σ) for the ”lifted”

family of forms on C^{ns}, corresponding to Σ, (†). By the transfer result, Lemma 2.31, the g_{n}^{r} on C^{ns}, used to define I_{italian}^{Σ,mobile}(p, C, γ_{p}^{j}, ϕ_{λ}), is obtained from the ”lifted” family of form on C^{ns} after removing all fixed point contributions. Therefore, as by (†), this lifted family has no fixed point contribution at p_{j}, we must have that;

I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) = I_{italian}^{Σ,mobile}(p, C, γ_{p}^{j}, ϕ_{λ}) (2) Combining (1),(2) and using Lemma 2.10, we have that; I_{italian}^{Σ}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) = I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ}) The result for this case now follows immediately from the definition

of I_{italian}^{Σ}(p_{j}, C^{ns}, \overline{ϕ_{λ}}).

Case 2. p ∈ Base(Σ). Then, by Definition 5.21, we have that; I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ}) = I_{italian}^{Σ,mobile}(p, C, γ_{p}^{j}, ϕ_{λ}) + 1 (1). In this case, we have that p_{j} ∈ Base(Σ) for the ”lifted” family of

forms on C^{ns}, corresponding to Σ, (†). Let I_{p_{j}} be the fixed point contribution for this family, as defined in Lemma 2.14. Then, by a similar argument to the above, we have that;

I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) = I_{p_{j}} + I_{italian}^{Σ,mobile}(p, C, γ_{p}^{j}, ϕ_{λ}) (2) Using Lemma 2.15, we have that; I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) = I_{p_{j}} + I_{italian}^{Σ}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) − 1 (3) Combining (1), (2), (3) gives that; I_{italian}^{Σ}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) = I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ})

Again, the result for this case follows immediately from the definition of I_{italian}^{Σ}(p_{j}, C^{ns}, \overline{ϕ_{λ}}).

□

As an easy consequence of the previous lemma, we have that;

Lemma 5.23. Let C ⊂ P^{w} be a projective algebraic curve. Let Σ be a linear system, having finite intersection with C. Then, if γ_{p}^{j} is a branch centred at p and ϕ_{λ} belongs to Σ;

I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ}) = Card(C ∩γ_{p}^{j} ∩ϕ_{λ′}) for λ′ ∈ V_{λ} generic in P ar_{Σ}.

I_{italian}^{Σ,mobile}(p, C, γ_{p}^{j}, ϕ_{λ}) = Card(C ∩ (γ_{p}^{j} \ p) ∩ ϕ_{λ′}) for λ′ ∈ V_{λ} generic in P ar_{Σ}.

where γ_{p}^{j} was given in Definition 5.15.

Proof. The first part of the lemma follows immediately from Lemma 5.22 and the Definition 5.15 of a branch. The second part follows from Definition 5.21 and the first part. □

We then reformulate the remaining results of Section 2 in terms of branches. The notation of Lemma 5.23 will be use for the remainder of this section.

Lemma 5.24. Non-Existence of Coincident Mobile Points along a Branch

Let C ⊂ P^{w} be a projective algebraic curve. Let Σ be a linear system, having finite intersection with C, such that p ∈ C \ Base(Σ). Then, if γ_{p}^{j} is a branch centred at p and ϕ_{λ} belongs to Σ;

I_{italian}(p, C, γ_{p}^{j}, ϕ_{λ}) = I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ}) = I_{italian}^{Σ,mobile}(p, C, γ_{p}^{j}, ϕ_{λ})

Proof. Let (C^{ns}, Φ) be a non-singular model of C, with presentation Φ_{Σ′}, such that Base(Σ′) is disjoint from Γ_{[Φ]}(x, p). Let \{\overline{ϕ_{λ}}\} be the ”lifted” family of algebraic forms on C^{ns} defined by Σ. By Lemma 5.22, we have that;

I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ}) = I_{italian}^{Σ}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) (1)

By Remarks 5.10, we have that;

I_{italian}(p, C, γ_{p}^{j}, ϕ_{λ}) = I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) (2)

Using the fact that p ∉ Base(Σ) and the hypotheses on Φ_{Σ′}, it follows that p_{j} ∉ Base(Σ), for the lifted system defined by Σ. Hence, we may apply Lemma 2.10, to obtain that;

I_{italian}^{Σ}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) = I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) (3)

The result follows by combining (1), (2) and (3) and using Definition 5.21.

□

Lemma 5.25. Branch Multiplicity at non-base points witnessed by transverse intersections along the branch

Let p ∈ C \ Base(Σ) and let γ_{p}^{j} be a branch centred at p. Then, if m = I_{italian}(p, C, γ_{p}^{j}, ϕ_{λ}), we can find λ′ ∈ V_{λ}, generic in P ar_{Σ}, and distinct {p_{1}, …, p_{m}} = C ∩ ϕ_{λ′} ∩ (γ_{p}^{j} \ p) such that the intersection of C with ϕ_{λ′} at each p_{i} is transverse for 1 ≤ i ≤ m.

Proof. By Lemmas 5.23 and 5.24, for λ′ ∈ V_{λ}, generic in P ar_{Σ}, the intersection C ∩ϕ_{λ′} ∩γ_{p}^{j} consists of m distinct points {p_{1}, …, p_{m}}. The condition on P ar_{Σ} that ϕ_{λ} passes through p defines a proper closed subset, hence we may assume these points are all distinct from p. Finally, the transversality result follows from, say Lemma 2.17, using the fact that {p_{1}, …, p_{m}} cannot lie inside Base(Σ).

□

Again, we have analogous results to Lemmas 5.24 and 5.25 for points in Base(Σ);

We first require the following;

Lemma 5.26. Let p ∈ C ∩ Base(Σ) and γ_{p}^{j} a branch centred at p. Then there exists an open subset U_{γ_{p}^{j}} ⊂ P ar_{Σ} and an integer I_{γ_{p}^{j}} ≥ 1 such that;

I_{italian}(p, C, γ_{p}^{j}, ϕ_{λ}) = I_{γ_{p}^{j}} for λ ∈ U_{γ_{p}^{j}}

and

I_{italian}(p, C, γ_{p}^{j}, ϕ_{λ}) ≥ I_{γ_{p}^{j}} for λ ∈ P ar_{Σ}

Proof. Let (C^{ns}, Φ) be a non-singular model with presentation Φ_{Σ′} such that Base(Σ′) is disjoint from Γ_{[Φ]}(x, p). Then, by the proof of Lemma 5.12, we have that, for λ ∈ P ar_{Σ};

I_{italian}(p, γ_{p}^{j}, C, ϕ_{λ}) = I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}})

By properties of Zariski structures;

W_{k} = \{λ ∈ P ar_{Σ} : I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) ≥ k\}

are definable and Zariski closed in P ar_{Σ}. The result then follows by taking I_{γ_{p_{j}}} = min_{λ∈P ar_{Σ}}I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) and using the fact that P ar_{Σ} is irreducible.

□

We can now formulate analogous results to Lemmas 5.24 and 5.25 for base points;

Lemma 5.27. Let p ∈ C ∩ Base(Σ) ∩ ϕ_{λ} and γ_{p}^{j} a branch centred at p. Then;

I_{italian}(p, γ_{p}^{j}, C, ϕ_{λ}) = I_{γ_{p}^{j}} + I_{italian}^{Σ}(p, γ_{p}^{j}, C, ϕ_{λ}) − 1

and

I_{italian}(p, γ_{p}^{j}, C, ϕ_{λ}) = I_{γ_{p}^{j}} + I_{italian}^{Σ,mobile}(p, γ_{p}^{j}, C, ϕ_{λ})

Proof. In order to prove the first part of the lemma, suppose that m = I_{italian}^{Σ}(p, γ_{p}^{j}, C, ϕ_{λ}). By Lemma 5.23, choosing λ′ ∈ V_{λ} generic in P ar_{Σ}, we can find {p_{1}, …, p_{m−1}} = C ∩ ϕ_{λ′} ∩ (V_{p} \ p), distinct from p, witnessing this multiplicity. Using the fact that {p_{1}, …, p_{m−1}} lie outside Base(Σ), we may apply Lemma 2.17 to obtain that the intersections at these points are transverse. By the previous Lemma 5.26, we have that I_{italian}(p, γ_{p}^{j}, C, ϕ_{λ′}) = I_{γ_{p}^{j}}. Now choose a nonsingular model (C^{ns}, Φ), with presentation Φ_{Σ′}, such that Base(Σ′) is disjoint from Γ_{[Φ]}(x, p). By definition 5.15 of a branch, we can find \{p′_{1}, …, p′_{m}\} = C^{ns} ∩ \overline{ϕ_{λ′}} ∩ (V_{p} \ p). By properties of specialisations, Base(Σ′) is also disjoint from this set. We then have that the intersections between C^{ns} and \overline{ϕ_{λ′}} are also transverse at these points and that I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ′}}) = I_{γ_{p}^{j}}. It then follows by summability of specialisation, see [6], that;

I_{italian}(p_{j}, C^{ns}, \overline{ϕ_{λ}}) = I_{γ_{p}^{j}} + (m − 1).

Again, using the presentation of (C^{ns}, Φ), we obtain that;

I_{italian}(p, γ_{p}^{j}, C, ϕ_{λ}) = I_{γ_{p}^{j}} + (m − 1).

Hence, the result follows. The second part of the lemma follows immediately from the Definition 5.21 of I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ}) at a base point. □

Lemma 5.28. Let p ∈ C ∩ Base(Σ) and let γ_{p}^{j} be a branch centred at p. Then, if m = I_{italian}^{Σ}(p, C, γ_{p}^{j}, ϕ_{λ}), we can find λ′ ∈ V_{λ}, generic in P ar_{Σ}, and distinct {p_{1}, …, p_{m−1}} = C ∩ ϕ_{λ′} ∩ (γ_{p}^{j} \ p) such that the intersection of C with ϕ_{λ′} at each p_{i} is transverse for 1 ≤ i ≤ m − 1. If m = I_{italian}^{Σ,mobile}(p, C, γ_{p}^{j}, ϕ_{λ}), then the same results for distinct {p_{1}, …, p_{m}} with the same properties.

Proof. The first part of the lemma is a straightforward consequence of Lemma 5.23, properties of infinitesimals, (to show that {p_{1}, …, p_{m−1}} lie outside Base(Σ)) and Lemma 2.17 (to obtain transversality). The second part of the lemma also follows from Lemma 5.23 and Lemma 2.17 (to obtain transversality). □

6. Cayley’s Classification of Singularities

The purpose of this section is to develop a theory of singularities for algebraic curves based on the work of Cayley. In order to make this theory rigorous, one first needs to find a method of parametrising the branches of an algebraic curve. This is the content of the following theorem;

Theorem 6.1. Analytic Representation of a Branch

Let C ⊂ P^{w} be a projective algebraic curve. Suppose that C is defined by equations {F_{1}(x_{1}, …, x_{w}), … F_{m}(x_{1}, …, x_{w})} in affine coordinates x_{i} = \frac{X_{i}}{X_{0}} . Let p ∈ C correspond to the point ¯0 in this coordinate system. Then there exist algebraic power series {x_{1}(t), …, x_{w}(t)} such that x_{1}(t) = … = x_{w}(t) = 0, F_{j}(x_{1}(t), …, x_{w}(t)) = 0 for 1 ≤ j ≤ m and with the property that, for any algebraic function F_{λ}(x_{1}, …, x_{w}); F_{λ}(x_{1}(t), …, x_{w}(t)) ≡ 0 iff F_{λ} vanishes on C.

Otherwise, F_{λ} has finite intersection with C and

ord_{t}F_{λ}(x_{1}(t), …, x_{w}(t)) = I_{italian}(p, γ_{p}^{j}, C, F_{λ}) (∗)

We refer to the power series as parametrising the branch γ_{p}^{j}.

Proof. We first prove the theorem in the case when C ⊂ P^{w} is a nonsingular projective algebraic curve. By Lemma 4.13, we can find a plane projective algebraic curve C_{1} ⊂ P^{2} such that {C, C_{1}} are birational and there exists a corresponding point p_{1} ∈ C_{1} such that p_{1} is non-singular. Let Φ_{Σ} and Ψ_{Σ′} be presentations such that Ψ_{Σ′} = Φ_{Σ}^{−1} as a birational map. Without loss of generality, we may assume that V_{Φ_{Σ}} = W_{Ψ_{Σ′}} ⊂ C and V_{Ψ_{Σ′}} = W_{Φ_{Σ}} ⊂ C_{1}. Moreover, we may assume that {p, p_{1}} lie in {V_{Φ_{Σ}}, V_{Ψ_{Σ′}}} and correspond to the origins of the affine coordinate systems (x_{1}, …, x_{w}) and (y_{1}, y_{2}). Let Σ′ = {ψ_{0}, ψ_{1}, …, ψ_{w}} and let (y_{1}(t), y_{2}(t)) be an analytic representation of p_{1} ∈ C_{1}, given by the inverse function theorem. We obtain an analytic representation of p ∈ C by the formula;

(x_{1}(t), …, x_{w}(t)) = (\frac{ψ_{1}}{ψ_{0}} (y_{1}(t), y_{2}(t)), …, \frac{ψ_{w}}{ψ_{0}} (y_{1}(t), y_{2}(t)))

First, note that as p ∉ Zero(ψ_{0}), we may assume that ψ_{0}(0, 0) ≠ 0. Hence, we can invert the power series ψ_{0}(y_{1}(t), y_{2}(t)). This clearly proves that x_{j}(t) is a formal power series in L[[t]]. That x_{j}(0) = 0 for 1 ≤ j ≤ w follows from the corresponding property for (y_{1}(t), y_{2}(t)) and the fact that p is situated at the origin of the coordinate system (x_{1}, …, x_{w}). Finally, we need to check that x_{j}(t) define algebraic power series. This follows obviously from the fact that ψ_{j} and ψ_{0} define algebraic functions. Now, suppose that {F_{1}, …, F_{m}} are defining equations for C. Let {F′_{1}, …, F′_{m}} be the corresponding equations written in homogeneous form for the variables {X_{0}, …, X_{w}}, where x_{j} = \frac{X_{j}}{X_{0}} . Let G(Y_{0}, Y_{1}, Y_{2}) be the defining equation for C_{1}. We can homogenise the power series representation of p_{1} ∈ C_{1} by (Y_{0}(t) : Y_{1}(t) : Y_{2}(t)) = (1 : y_{1}(t) : y_{2}(t)). Then we must have that G(1 : y_{1}(t) : y_{2}(t)) = 0. Now F′_{k}(ψ_{0}, …, ψ_{w}) vanishes identically on C_{1}, hence, by the projective Nullstellensatz, there exists a homogeneous H_{k}(Y_{0}, Y_{1}, Y_{2}) such that;

F′_{k}(ψ_{0}, …, ψ_{w}) = H_{k}G It follows that;

F′_{k}(ψ_{0}(1 : y_{1}(t) : y_{2}(t)), …, ψ_{w}(1 : y_{1}(t) : y_{2}(t))) ≡ 0

therefore;

F′_{k}(1 : \frac{ψ_{1}(y_{1}(t),y_{2}(t))}{ψ_{0}(y_{1}(t),y_{2}(t))} : …: \frac{ψ_{w}(y_{1}(t),y_{2}(t))}{ψ_{0}(y_{1}(t),y_{2}(t))}) ≡ 0

which gives;

F_{k}(x_{1}(t), …, x_{w}(t)) ≡ 0

as required. The property that an algebraic function F_{λ} vanishes on (x_{1}(t), …, x_{w}(t)) iff it vanishes on C can be proved in a similar way to the above argument, invoking Theorem 5.1 of the paper [6]. Alternatively, it can be proved directly, using the fact that, as (x_{1}(t), …, x_{w}(t)) define algebraic power series, (y_{1} − x_{1}(t), …, y_{w} − x_{w}(t)) defines the equation of a curve C′ on some etale cover i : (A_{et}^{w}, (¯0)^{lift}) → (A^{w}, (¯0)) such that i(C′) ⊂ C. If F_{λ} vanishes on C′, then it must vanish on an open subset U of C, hence as F_{λ} is closed, must vanish on all of C as required. Finally, we need to check the property (∗). Suppose that F_{λ} is an algebraic function with m = ord_{t}F_{λ}(x_{1}(t), …, x_{w}(t)), passing through p. Choose Σ_{1} containing F_{λ} such that Σ_{1} has finite intersection with C and p ∉ Base(Σ_{1}). It follows, using Lemma 2.12, that I_{italian}(p, C, F_{λ}) = I_{italian}^{Σ_{1}}(p, C, F_{λ}), (†). Using Lemma 2.31, we can transfer the system Σ_{1} to a system on C_{1}. Let G_{λ} be the corresponding algebraic curve to the algebraic form F_{λ}. We must have that p_{1} ∉ Base(Σ_{1}), otherwise, as p_{1} belongs to the canonical set V_{Ψ_{Σ′}}, we would have that p belongs to Base(Σ_{1}) as well. Hence, using Lemma 2.12 again, we must have that I_{italian}(p_{1}, C_{1}, G_{λ}) = I_{italian}^{Σ_{1}}(p_{1}, C_{1}, G_{λ}), (††). By direct calculation, we have that;

G_{λ}(y_{1}(t), y_{2}(t)) = ψ_{0}^{r}(y_{1}(t), y_{2}(t))F_{λ}(x_{1}(t), …, x_{w}(t))

for some r ≤ 0. As ord_{t}ψ_{0}(y_{1}(t), y_{2}(t)) = 0, we have that;

ord_{t}G_{λ}(y_{1}(t), y_{2}(t)) = ord_{t}F_{λ}(x_{1}(t), …, x_{w}(t)) = m

Now, by Theorem 5.1 of the paper [6] and (††), it follows that; I_{italian}(p_{1}, C_{1}, G_{λ}) = I_{italian}^{Σ_{1}}(p_{1}, C_{1}, G_{λ}) = m

Now, using Lemma 2.31 and the fact that {p, p_{1}} lie in the canonical sets V_{Φ_{Σ}} and W_{Φ_{Σ}}, we must have that I_{italian}^{Σ_{1}}(p, C, F_{λ}) = m as well. Hence, by (†), it follows that I_{italian}(p, C, F_{λ}) = m. As C is a nonsingular model of itself, this proves the claim (∗) in this special case.

We now assume that C ⊂ P^{w} is any projective algebraic curve. Suppose that C^{ns} ⊂ P^{w′} is a non-singular model of C with birational morphism Φ_{Σ′} : C^{ns} → C such that the branch γ_{p}^{j} corresponds to V_{p_{j}} in the fibre Γ_{[Φ]}(x, p), disjoint from Base(Σ′). As before, we may assume that {p, p_{j}} correspond to the origins of the coordinate systems (x_{1}, …, x_{w}) and (y_{1}, …, y_{w′}). Let Σ′ = {ϕ_{0}, …, ϕ_{w}}. By the previous argument, we can find an analytic representation (y_{1}(t), …, y_{w′}(t)) of p_{j} in C^{ns}, with the properties given in the statement of the theorem. As before, we obtain an analytic representation of the corresponding p ∈ C, by the formula;

(x_{1}(t), …, x_{w}(t)) = (\frac{ϕ_{1}}{ϕ_{0}} (y_{1}(t), …, y_{w′}(t)), …, \frac{ϕ_{w}}{ϕ_{0}} (y_{1}(t), …, y_{w′}(t)))

One checks that this has the required properties up to the property (∗) by a direct imitation of the proof above, with the minor modification that the projective Nullstellensatz for C^{ns} gives that, if {G_{1}, …, G_{k}} are defining equations for C^{ns}, then, if F vanishes on C^{ns}, there must exist homogeneous {H_{1}, …, H_{k}} such that F = H_{1}G_{1}+…+H_{k}G_{k}. Alternatively, one can refer the parametrisation to a non-singular point of a plane projective curve, in which case the argument up to (∗) is identical.

We now verify the property (∗). Suppose that F_{λ} is an algebraic function with ord_{t}F_{λ}(x_{1}(t), …, x_{w}(t)) = m. Let \overline{F}_{λ} be the corresponding function on C^{ns}, obtained from the presentation Φ_{Σ′}. By Lemmas 5.11 and 5.12;

I_{italian}(p, γ_{p}^{j}, C, F_{λ}) = I_{italian}(p_{j}, C^{ns}, \overline{F_{λ}}) (∗∗)

We claim that ord_{t}\overline{F}_{λ}(y_{1}(t), …, y_{w′}(t)) = m. This follows by repeating the argument given above. By the properties of (y_{1}(t), …, y_{w′}(t)) and the result verified in the case of a non-singular curve, we obtain immediately that I_{italian}(p_{j}, C^{ns}, \overline{F_{λ}}) = m. Combined with (∗∗), this gives the required result.

□

Using the analytic representation, we obtain the following classification of singularities due to Cayley;

Theorem 6.2. Cayley’s Classification of Singularities

Let C ⊂ P^{w} be a projective algebraic curve which is not contained in any hyperplane section. Let γ_{p}^{j} be a branch of the algebraic curve centred at p. Then we can assign a sequence of non-negative integers (α_{0}, α_{1}, …, α_{w−1}), called the character of the branch, which has the following property;

Let Σ be the system of hyperplanes passing through p. Then there exists a filtration of Σ into subsystems of hyperplanes;

Σ_{w−1} ⊂ Σ_{w−2} ⊂ … ⊂ Σ_{1} ⊂ Σ_{0} = Σ

with dim(Σ_{i}) = (w −1)−i, such that, for any hyperplane H_{λ} passing through p, we have that;

H_{λ} ∈ Σ_{i} iff I_{italian}(p, γ_{p}^{j}, C, H_{λ}) ≥ α_{0} + α_{1} + … + α_{i}. (†)

Moreover, these are the only multiplicities which occur.

Proof. Without loss of generality, we may assume that p is situated at the origin of the coordinate system (x_{1}, …, x_{w}). By the previous theorem, we can find an analytic parameterisation of the branch γ_{p}^{j} of the form;

x_{k}(t) = a_{k,1}t + a_{k,2}t^{2} + … + a_{k,n}t^{n} + …, for 1 ≤ k ≤ w

Let ∑_{k=1}^{w} λ_{k}x_{k} = 0 be the equation of a hyperplane H_{λ} passing through p. Then, we can write H_{λ}(x_{1}(t), …, x_{k}(t)) in the form;

(∑_{k=1}^{w} λ_{k}a_{k,1})t + … + (∑_{k=1}^{w} λ_{k}a_{k,n})t^{n} + O(t^{n}) (*)

Let ā_{n} = (a_{1,n}, …, a_{k,n}, …, a_{w,n}). We claim that we can find a sequence (ā_{m_{1}}, …, ā_{m_{w}}), for m_{1} < … < m_{i} < … < m_{w}, such that; (i). {ā_{m_{1}}, …, ā_{m_{w}}} is linearly independent

(ii). V_{m_{i}} =< ā_{1}, …, ā_{m_{i}} >=< ā_{m_{1}}, …, ā_{m_{i}} >, for 1 ≤ i ≤ w

(iii). If there exist {n_{1}, …, n_{i}} for n_{1} < … < n_{i} with n_{1} ≤ m_{1}, …, n_{i} ≤ m_{i} such that

V_{m_{i}} =< ā_{n_{1}}, …, ā_{n_{i}} >,

then n_{1} = m_{1}, …, n_{i} = m_{i}.

(iv). V_{k} = V_{m_{i}} for m_{i} ≤ k < m_{i+1}.

The first three properties may be proved by induction on i.For i = 1, choose the first non-zero vector ā_{m_{1}}. For the inductive step, assume we have found {ā_{m_{1}}, …, ā_{m_{i}}} with the required properties (i) − (iii). We claim that there exists ā_{m_{i+1}}, with m_{i+1} > m_{i}, such that ā_{m_{i+1}} ∉ V_{m_{i}}, (∗∗). Suppose not. As i < w, the condition;

⋀_{s=1}^{i}(∑_{k=1}^{w} λ_{k}a_{k,m_{s}} = 0)

defines a non-empty plane P in the dimension w−1 parameter space P ar_{H} of hyperplanes passing through p. Choosing λ ∈ P and using (∗), it follows that ord_{t}H_{λ}(x_{1}(t), …, x_{w}(t)) ≥ n for all n > 0. Therefore, H_{λ}(x_{1}(t), …, x_{w}(t)) ≡ 0 and, by Theorem 6.1, H_{λ} must contain C. This contradicts the assumption that C is not contained in any hyperplane section. Using (∗∗), choose m_{i+1} minimal such that ā_{m_{i+1}} ∉ V_{m_{i}}. Properties (i) and (ii) are trivial to verify. For (iii), assume that {ā_{n_{1}}, …, ā_{n_{i+1}}} are as given in the hypotheses. Then, the sequence must form a linearly independent set. Hence, we must have that V_{m_{i}} =< ā_{n_{1}}, …, ā_{n_{i}} >. By the induction hypothesis, n_{1} = m_{1}, …, n_{i} = m_{i}. Then, ā_{n_{i+1}} ∉ V_{m_{i}}, Now, by minimality of m_{i+1}, we also have that ā_{m_{i+1}} = ā_{n_{i+1}} as required.

Property (iv) follows easily from properties (i) − (iii). We clearly have that V_{m_{i}} ⊆ V_{k} ⊆ V_{m_{i+1}}, for m_{i} ≤ k < m_{i+1}. If V_{k} ≠ V_{m_{i}}, then, by (i), (ii), V_{k} = V_{m_{i+1}}. This clearly contradicts (iii).

Now, define;

Σ_{i} = {λ ∈ P ar_{H,p} : ⋀_{s=1}^{i}(∑_{k=1}^{w} λ_{k}a_{k,m_{s}} = 0)}, for, 1 ≤ i ≤ w − 1, Then, we obtain a filtration;

Σ_{w−1} ⊂ … ⊂ Σ_{i} ⊂ … ⊂ Σ_{0} = Σ

with dim(Σ_{i}) = (w−1)−i as in the statement of the theorem. Define;

α_{0} = m_{1} and α_{i} = m_{i+1} − m_{i} for 1 ≤ i ≤ w − 1.

We need to verify the property (†). Suppose that H_{λ} ∈ Σ_{i}, for i ≥ 1. Then H_{λ} contains the plane V_{m_{i}} spanned by {ā_{m_{1}}, …, ā_{m_{i}}}. Hence, by (iv), it contains the plane V_{k} for k < m_{i+1}. Then, by (∗), ord_{t}(H_{λ}(x_{1}(t), …, x_{w}(t))) ≥ m_{i+1} and ,by Theorem 6.1,

I_{italian}(p, γ_{p}^{j}, C, H_{λ}) ≥ m_{i+1} = α_{0} + … + α_{i}.

Conversely, suppose that

ord_{t}H_{λ}(x_{1}(t), …, x_{w}(t)) ≥ α_{0} + … α_{i} = m_{i+1}, for some i ≥ 1.

Then H_{λ} contains the plane V_{k} for k < m_{i+1}. In particular, it contains the plane V_{m_{i}}. Hence H_{λ} ∈ Σ_{i}. The remaining case amounts to showing that I_{italian}(p, γ_{p}^{j}, C, H_{λ}) ≥ α_{0} for any hyperplane H_{λ} passing through p. This follows immediately from (∗), Theorem 6.1 and the fact that ā_{m_{1}} was the first non-zero vector in the sequence {ā_{n} : n < ω}. The remark made after the property (†) follows immediately from the property (iv) of the sequence {ā_{m_{1}}, …, ā_{m_{w}}}.

□

Definition 6.3. In accordance with the Italian terminology, we refer to α_{0} as the order of the branch, α_{j} as the j’th range of the branch, for 1 ≤ j ≤ w − 2, and α_{w−1} as the final range or class of the branch. We define < ā_{m_{1}}, …, ā_{m_{k}} > to be the k’th osculatory plane at p for 1 ≤ k ≤ w − 1. We also define the w − 1’th osculatory plane to be the osculatory plane. We define the tuple (α_{0}, …, α_{w−1}) to be the character of the branch. Cayley referred to branches of order 1 as linear and superlinear otherwise. He referred to branches having a character of the form (1, 1 …, 1) as ordinary. The Italian geometers refer to a simple point, which is the origin of an ordinary branch, as an ordinary simple point. Note that for a simple (equivalently non-singular) point, the 1’st osculatory plane is the same as the tangent line. We define a node of a plane curve to be the origin of at most 2 ordinary branches with distinct tangent lines, this definition was used in Theorem 4.16.

We also used, in Theorem 4.16, the fact that a generic point of an algebraic curve is an ordinary simple point, (this is not true when the field has non-zero characteristic, see the final section) a rigorous proof of this result requires duality arguments, we postpone this proof for another occasion.

We have the following important results on the projection of a branch, see Section 4 for the relevant definitions.

Theorem 6.4. Let C ⊂ P^{w} be a projective algebraic curve, as defined in the previous theorems of this section, and γ_{O} a branch centred at O with character (α_{0}, …, α_{w−1}). Let P be chosen generically in P^{w}, then the projection pr_{P} (γ_{O}) has character;

(α_{0}, …, α_{w−2})

If P is situated generically on the osculatory plane, then pr_{P} (γ_{O}) has character;

(α_{0}, …, α_{w−2} + α_{w−1})

More generally, if P is situated on the k’th osculatory plane for 1 ≤ k ≤ w − 2, and not on an osculatory plane of lower order, then the projection pr_{P} (γ_{O}) has character;

(α_{0}, …, α_{k−2}, α_{k−1} + α_{k}, α_{k+1}, …, α_{w−1}).

Proof. First note that, by Lemma 4.9, if P is situated generically on the k’th-osculatory plane for any k ≥ 1, and H′ is any hyperplane not containing P, the projection pr_{P} is generally biunivocal. Hence, by Lemma 5.7, the projection pr_{P} (γ_{O}) is well defined. We first claim that for any hyperplane H_{λ} in P^{w−1};

I_{italian}(O, γ_{O}, C, pr_{P}^{−1}(H_{λ})) = I_{italian}(pr_{P} (O), pr_{P} (γ_{O}), pr_{P} (C), H_{λ}) (∗)

This follows by using the proof of Lemma 4.13 and the fact that the multiplicity is calculated at a branch, to replace the use of biunivocity. Now, if P is situated in generic position in P^{w}, then {pr_{P} (ā_{m_{1}}), …, pr_{P} (ā_{m_{w−1}})} forms a linearly independent sequence passing through pr_{P} (O) ∈ P^{w−1} and, for any hyperplane H_{λ} ⊂ P^{w−1}, we have that, for i ≤ w − 2;

A THEORY OF BRANCHES FOR ALGEBRAIC CURVES 79

< pr_{P} (ā_{m_{1}}), …, pr_{P} (ā_{m_{i}}) >⊂ H_{λ} iff < ā_{m_{1}}, …, ā_{m_{i}} >⊂< P, H_{λ} > It follows by (∗) and Theorem 6.2 that; I_{italian}(pr_{P} (O), pr_{P} (γ_{O}), pr_{P} (C), H_{λ}) ≥ α_{0} + … α_{i} iff < pr_{P} (ā_{m_{1}}), …, pr_{P} (ā_{m_{i}}) >⊂ H_{λ} (i ≤ w − 2) Hence, by Theorem 6.2 again, we have that the branch pr_{P} (γ_{O}) has

character (α_{0}, …, α_{w−2}).

If P is situated generically on the k’th oscillatory plane, but not on an oscillatory plane of lower order, then, the sequence {pr_{P} (ā_{m_{1}}), …, pr_{P} (ā_{m_{k−1}}), pr_{P} (ā_{m_{k+1}}), …, pr_{P} (ā_{m_{w−1}})} forms a linearly independent set, and, for 1 ≤ i ≤ k − 2;

< pr_{P} (ā_{m_{1}}), …, pr_{P} (ā_{m_{i}}) >⊂ H_{λ} iff < ā_{m_{1}}, …, ā_{m_{i}} >⊂< P, H_{λ} > whereas; < pr_{P} (ā_{m_{1}}), …, pr_{P} (ā_{m_{k−1}}) >⊂ H_{λ} iff < ā_{m_{1}}, …, ā_{m_{k}} >⊂< P, H_{λ} > and, for 1 ≤ j ≤ (w − 2 − k); < pr_{P} (ā_{m_{1}}), …, pr_{P} (ā_{m_{k−1}}), pr_{P} (ā_{m_{k+1}}), …, pr_{P} (ā_{m_{k+j}}) >⊂ H_{λ} iff < ā_{m_{1}}, …, ā_{m_{k+j}} >⊂< P, H_{λ} > The rest of the theorem then follows immediately by the same argu-

ment as given above.

□

We also have the following important consequence of Theorem 6.1. Theorem 6.5. Linearity of Multiplicity at a Branch

Let C ⊂ P^{w} be a projective algebraic curve and γ_{p}^{j} a branch of C, centred at p. Let Σ be an independent system having finite intersection with C. Then, for any k ≥ 1, the condition;

{λ ∈ P ar_{Σ} : I_{italian}(p, γ_{p}^{j}, C, F_{λ}) ≥ k} (∗)

is linear and definable.

Proof. That (∗) is definable follows from the definition of multiplicity at a branch and elementary facts about Zariski structures. Moreover, (∗) is definable over the parameters of C and the point p. In order to prove linearity, suppose that F_{λ} and F_{μ} belong to Σ and satisfy (∗). Let (x_{1}(t), …, x_{w}(t)) be a parameterisation of the branch γ_{p}^{j} as given by Theorem 6.1. Then;

k ≤ min{ord_{t}F_{λ}(x_{1}(t), …, x_{w}(t)), ord_{t}F_{μ}(x_{1}(t), …, x_{w}(t))}

Then, for any constant c;

(F_{λ}+cF_{μ})(x_{1}(t), …, x_{w}(t)) = F_{λ}(x_{1}(t), …, x_{w}(t))+cF_{μ}(x_{1}(t), …, x_{w}(t))

Hence,

ord_{t}(F_{λ} + cF_{μ})(x_{1}(t), …, x_{w}(t)) ≥ k

as well. This shows that the pencil of curves generated by {F_{λ}, F_{μ}} satisfies (∗). Let W be the closed projective subvariety of P ar_{Σ} defined by the condition (∗). Then, W has the property that for any {a, b} ⊂ W, l_{ab} ⊂ W. It follows easily that W defines a plane H in P ar_{Σ} as required. (Use the fact that for any tuple {a_{1}, …, a_{n}} in W, the plane H_{a_{1},...,a_{n}} ⊂ W and a dimension argument)

□

Remarks 6.6. Note that, given Σ of dimension n as in the statement of the theorem, we can, using the above theorem, find a sequence {β_{0}, …, β_{n−1}} and a filtration;

Σ_{n−1} ⊂ … ⊂ Σ_{i} ⊂ … ⊂ Σ_{0} ⊂ Σ

with dim(Σ_{i}) = (n − 1) − i such that;

I_{italian}(p, γ_{p}^{j}, C, F_{λ}) ≥ β_{0} + … + β_{i} iff F_{λ} ∈ Σ_{i}

and these are the only multiplicities which can occur. Note also that, as an easy consequence of the theorem, given any tuple (β_{0}, …, β_{n−1}) and Σ as above;

{x ∈ C : x has character (β_{0}, …, β_{n−1}) with respect to Σ}

is constructible and defined over the field of definition of C and Σ. In particular, it follows from the previous Definition 6.3, in characteristic 0, that there exist only finitely many points on C which are the origins of non-ordinary branches.

We have the following important characterisation of multiplicity at a branch;

Theorem 6.7. Multiplicity at a Branch as a Specialised Condition

Let C and Σ be as in the statement of Theorem 6.5 and the following remark. Fix independent generic points (over L) {p_{0j}, …, p_{ij}} in γ_{p}^{j}. Then the system Σ_{i} may be obtained by specialising the condition;

{λ ∈ P ar_{Σ} : F_{λ} = 0 passes through {p_{0j}, …, p_{ij}}}.

That is, if F_{λ} ∈ Σ_{i}, there exists λ′ ∈ V_{λ} such that F_{λ′} passes through {p_{0j}, …, p_{ij}}, while, if F_{λ′} passes through {p_{0j}, …, p_{ij}}, then its specialisation F_{λ} belongs to Σ_{i}.

Proof. We will first assume that the curve C is non-singular, (hence, by Lemma 5.4, there exists a single branch at p). Consider the cover F_{i} ⊂ (C \ p) × P ar_{Σ} given by;

F_{i+1}(x, λ) ≡ (x ∈ C ∩ F_{λ}) ∧ (F_{λ} ∈ Σ_{i}) (i ≥ 0)

For generic q ∈ C, the fibre F_{0}(q, y) has dimension n − 1 − (i + 1). Hence, we obtain an open subset U ⊂ C such that F_{i+1} ⊂ (U \p)×P ar_{Σ} is regular with fibre dimension n − 1 − (i + 1). Let F̄_{i+1} be the closure of F_{i+1} in U × P ar_{Σ}. We claim that Σ_{i+1} is defined by the fibre F̄_{i+1}(p, y). First observe that, as p has codimension 1 in U, dim(F̄_{i+1}(p, y)) ≤ n−1−(i+1). As p is non-singular, each component of the fibre F̄_{i+1}(p, y) has dimension at least n − 1 − (i + 1). Suppose that F̄_{i+1}(p, λ) holds, then, by regularity of p for the cover F̄_{i+1}, given p′ ∈ U ∩ V_{p} generic, we can find λ′ ∈ V_{λ} such that F_{i+1}(p′, λ′), that is F_{λ′} passes through p′ and F_{λ′} belongs to Σ_{i}. By definition of Σ_{i}, we have that I_{italian}(p, C, F_{λ′}) ≥ β_{0} + … + β_{i}. As p′ is distinct from p, by summability of specialisation, see the paper [6], we must have that I_{italian}(p, C, F_{λ}) ≥ β_{0} + … + β_{i} + 1. Therefore, in fact, by the above theorem, I_{italian}(p, C, F_{λ}) ≥ β_{0} + … + β_{i+1} and F_{λ} belongs to Σ_{i+1}. By dimension considerations, it follows that F̄_{i+1}(p, y) defines the system Σ_{i+1} as required.

We now prove one direction of the theorem by induction on i ≥ 0. Suppose that {p_{0j}, …, p_{i+1,j}} are given independent generic points in C ∩ V_{p}. By the above, if F_{λ} belongs to Σ_{i+1}, then there exists λ′ ∈ V_{λ} such that F_{λ′} belongs to Σ_{i} and passes through p_{i+1,j}. Moreover, as all the covers F_{i} are defined over the field of definition L of C, we may take λ′ to lie in the field L_{1} = L(p_{i+1,j})^{alg}. Hence, F_{λ′} does not pass through any of the other independent generic points {p_{0j}, …, p_{ij}}. Let L_{2} = L(p_{0j}, …, p_{ij})^{alg}. As dim(p_{kj}/L) = 1, for 1 ≤ k ≤ i, we may, without loss of generality, assume that L_{2} is linearly disjoint from L_{1} over L. Hence, by the amalgamation property for the universal specialisation, see the paper [7], we have that {p_{0j}, …, p_{ij}} still belong to V_{p} ∩ C when taking P (L_{1}) as the standard model. Now, we consider the subsystem Σ′ ⊂ Σ defined by;

Σ′ = {F_{λ} : F_{λ} passes through p_{i+1,j}}

As p_{i+1,j} was chosen to be generic, it cannot be a base point for any of the subsystems;

Σ_{n−1} ⊂ … ⊂ Σ_{i} ⊂ … ⊂ Σ_{0} ⊂ Σ

Hence, we obtain a corresponding filtration;

Σ_{n−2} ∩ Σ′ ⊂ … ⊂ Σ_{i} ∩ Σ′ ⊂ … ⊂ Σ_{0} ∩ Σ′ ⊂ Σ′

with the properties in Remarks 6.6. We now apply the induction hypothesis to F_{λ′} ∈ Σ_{i} ∩ Σ′. We can find λ′′ ∈ V_{λ′} such that F_{λ′′} passes through {p_{0j}, …, p_{ij}} and F_{λ′′} belongs to Σ′. Hence F_{λ′′} passes through {p_{0j}, …, p_{i+1,j}}. Finally, note that λ′′ ∈ V_{λ}, if one considers P (L) rather than P (L_{1}) as the standard model. Hence, one direction of the theorem is proved.

The converse direction may also be proved by induction on i ≥ 0. Suppose that F_{λ′′} passes through independent generic points {p_{0j}, …, p_{i+1,j}} in γ_{p}^{j}. As before, we may consider p_{i+1,j} as belonging to the standard model P (L_{1}) and {p_{0j}, …, p_{ij}} as belonging to V_{p} ∩ C, relative to P (L_{1}). We again consider the subsystem Σ′ ⊂ Σ as defined above. Let λ′ be the specialisation of λ′′ relative to P (L_{1}). Then, F_{λ′} belongs to Σ′, as Σ′ is defined over p_{i+1,j}. Moreover, by the inductive hypothesis applied to Σ′, F_{λ′} also belongs to Σ_{i}. We now apply the argument at the beginning of this proof, with P (L) as the standard model, to obtain that F_{λ} belongs to Σ_{i+1}, where λ is the specialisation of λ′, relative to P (L). Clearly λ′′ specialises to λ, hence the converse direction is proved.

We still need to consider the case for arbitrary C ⊂ P^{w}. Let C^{ns} ⊂ P^{w′} be a non-singular model of C and suppose that the presentation Φ_{Σ′} of (C^{ns}, Φ^{ns}) has Base(Σ′) disjoint from the fibre Γ_{[Φ^{ns}]}(y, p). Let γ_{p}^{j} correspond to the infinitesimal neighborhood V_{p_{j}} of p_{j} in Γ_{[Φ^{ns}]}(y, p) and let \{\overline{F}_{λ}\} be the system Σ of lifted forms on C^{ns} corresponding to the space of forms {F_{λ}} in Σ. By Lemma 5.12, it follows that, for λ ∈ P ar_{Σ}, I_{italian}(p, γ_{p}^{j}, C, F_{λ}) = I_{italian}(p_{j}, C^{ns}, \overline{F_{λ}}), hence the character (β_{0}, …, β_{n}) of the branch γ_{p}^{j} with respect to the system Σ is the same as the character of the branch γ_{p_{j}} with respect to the lifted system Σ. Moreover, we obtain the same filtration of P ar_{Σ}, as given in Remark 6.6, for both systems with respect to the branches {γ_{p}^{j}, γ_{p_{j}}} (†). Now, suppose that we are given independent generic points {p_{0j}, …, p_{ij}} in γ_{p}^{j}. Then, by definition of γ_{p}^{j}, we can find corresponding independent generic points {p′_{0j}, …, p′_{ij}} in γ_{p_{j}}. Now suppose that F_{λ} ∈ Σ_{i}, then the corresponding \overline{F_{λ}} belongs to Σ_{i}. By the proof of the above theorem for non-singular curves, we can find λ′ ∈ V_{λ} such that \overline{F_{λ′}} passes through {p′_{0j}, …, p′_{ij}}. Then, by definition, the corresponding F_{λ} passes through {p_{0j}, …, p_{ij}}. Conversely, suppose that we can find λ′ ∈ V_{λ} such that F_{λ′} passes through {p_{0j}, …, p_{ij}}. Then the corresponding \overline{F_{λ′}} passes through \{p′_{0j}, …, p′_{ij}\}. By the proof for non-singular curves, the specialisation \overline{F_{λ}} belongs to Σ_{i}. Hence, by the observation (†) above, the corresponding F_{λ} belongs to Σ_{i} as well. The theorem is then proved. □

Remarks 6.8. One can give a slightly more geometric interpretation of the preceding theorem as follows;

Consider the cover F ⊂ C^{i+1} × P ar_{Σ} given by;

F (p_{1}, …, p_{i+1}, λ) ≡ {p_{1}, …, p_{i+1}} ⊂ C ∩ F_{λ} = 0

Generically, the cover F over C^{i+1} has fibre dimension n − (i + 1). For a tuple (p, …, p) ∈ Δ^{i+1}, the dimension of the fibre F (p, …, p) is n−1. By the above, Σ_{i} ⊂ F (p, …, p), which has dimension n−(i+1), is regular for the cover, in the sense of the above theorem.

The theorem may be construed as a generalisation of an intuitive notion of tangency. i + 1 independent generic points on the branch γ_{p}^{j} determine a projective plane H_{i} of dimension i. As these i + 1 points converge independently to p, the plane H_{i} converges to the i’th osculatory plane at p. As with the proofs we have given of many of the original arguments in [10], the method using infinitesimals in fact reverses this type of thinking in favour of a more visual approach. In this case, we have shown that, by moving the i’th osculatory plane away from p, we can cut the branch γ_{p}^{j} in i + 1 independent generic points.

The theorem also provides an effective method of computing osculatory planes at a branch γ_{p}^{j} for p ∈ C.

We now require the following lemma;

Lemma 6.9. Let F_{λ′} have finite intersection with C, where the parameter λ′ is taken inside the non-standard model P (K). Then there exists a maximally independent set {p_{0j}, …, p_{ij}} of generic intersections (over L) inside γ_{p}^{j}.

Proof. Let W be the finite set of intersections inside γ_{p}^{j} of F_{λ′} with C. As C is strongly minimal, if dim(W/L) = i+1, then there exists a basis {p_{0j}, …, p_{ij}} of W over L. In particular, we have that {p_{0j}, …, p_{ij}} are generically independent points of C and are maximally independent in W. □

We finish this paper with the following theorem;

Theorem 6.10. Intersections along a Branch

Let hypotheses be as in the previous theorem. Let i be maximal such that F_{λ′} belongs to Σ_{i} and suppose that F_{λ′} intersects γ_{p}^{j} in the maximally independent set of generic points {p_{i+1,j}, …, p_{i+r,j}} over L, where r ≤ (n − 1) − i. Then, if the branch γ_{p}^{j} has character (β_{0}, …, β_{n}) with respect to the system Σ, F_{λ′} intersects (γ_{p}^{j} \p) in at least β_{i+1}+…+β_{i+r} points, counted with multiplicity.

Proof. We assume first that C is non-singular. Let F_{λ} be the specialisation of F_{λ′} relative to the standard model P (L). Then by Theorem 6.7, F_{λ} belongs to Σ_{i+r} (replace the system Σ in Theorem 6.7 by the system Σ_{i}.) Hence, I_{italian}(p, C, F_{λ}) ≥ β_{0} + … + β_{i+r}, whereas I_{italian}(p, C, F_{λ′}) = β_{0} + … + β_{i}. It follows immediately, by summability of specialisation, see the paper [6], that the total multiplicity of intersections of F_{λ′} with C inside the branch (γ_{p}^{j} \ p) is at least;

(β_{0} + … + β_{i+r}) − (β_{0} + … + β_{i}) = β_{i+1} + … + β_{i+r}

as required. If C is singular, let (C^{ns}, Φ^{ns}) be a non-singular model, with a presentation Φ_{Σ′} such that Base(Σ′) is disjoint from Γ_{[Φ]}(x, p). Then, given the maximally independent set of generic points {p_{i+1,j}, …, p_{i+r,j}}, in γ_{p}^{j}, for the intersection of C with F_{λ′}, we obtain a maximally independent set for the intersection \overline{F}_{λ′} ∩ C ∩ V_{p}. By the above, and the fact that the character of the branch γ_{p}^{j} with respect to \{F_{λ}\} equals the character of the branch γ_{p_{j}} with respect to \overline{F}_{λ}, we obtain that \overline{F}_{λ′} intersects the branch (γ_{p_{j}} \p_{j}) in at least β_{i+1}+…+β_{i+r} points with multiplicity. Hence, using for example Lemma 5.12, and the fact that Base(Σ′) is disjoint from γ_{p_{j}}, we obtain that F_{λ′} intersects (γ_{p}^{j} \ p) in at least β_{i+1} + … + β_{i+r} points with multiplicity as well. □

Remarks 6.11. Note that, in the statement of the theorem, one cannot obtain that F_{λ′} intersects the branch (γ_{p}^{j} \ p) in exactly β_{i+1} + … + β_{i+r} points, with multiplicity. For example, consider the algebraic curve C given in affine coordinates by x^{3} − y^{2} = 0. At (0, 0), this has a cusp singularity with character (2, 1). The tangent line or 1’st osculatory plane, is given by y = 0. The line y −ϵ = 0, where ϵ is an infinitesimal, cuts the branch of C at (0, 0) in exactly 3 = 2 + 1 points. However, the total transcendence degree (over L) of these points is clearly 1. Neither can one obtain that F_{λ′} intersects the branch (γ_{p}^{j} \ p) transversely. For example, consider the algebraic curve C given in affine coordinates by y−x^{2} = 0. Let Σ be the 2-dimensional system consisting of (projective) lines. As (0, 0) is an ordinary simple point of C, it has character (1, 1), which is also the character of (0, 0) with respect to the system Σ. Again, the tangent line or 1’st osculatory plane, is given by y = 0. The line y = (2ϵ)x − ϵ^{2} cuts the branch of C at (0, 0) in exactly one point (ϵ, ϵ^{2}), with multiplicity 2, and specialises to y = 0.

Remarks 6.12. The above theorem is critical in calculations relating to duality. We save this point of view for another occasion.

7. Some Remarks on Frobenius

When the field has non-zero characteristic, many of the above arguments are complicated by the Frobenius morphism. However, we take the point of view that this is an exception rather than a general rule, hence the results are true if we exclude unusual cases. We will consider each of the previous sections separately and point out where to make these modifications. We briefly remind the reader that given, algebraic curves C_{1} and C_{2}, by a generally biunivocal map, denoted for this section only using the repetition of notation, ϕ : C_{1} ↭ C_{2}, we mean a morphism ϕ, defined on an open subset U ⊂ C_{1}, such that ϕ defines a bijective correspondence between U and an open subset V of C_{2}. In characteristic 0, a generally biunivocal map is birational, in the sense of Definition 1.19. However, this is not true when the field has non-zero characteristic, Frobenius being a counterexample.

Section 1. The results of this section hold in arbitrary characteristic.

Section 2. We encounter the first problem in Theorem 2.3, the proof of which depends on Lemma 2.10. Unfortunately, Lemma 2.10 is not true in arbitrary characteristic. However, as we will explain below, Lemma 2.10 is true for a linear system Σ which defines a birational morphism Φ_{Σ} on C. As this was assumed in Theorem 2.3, its proof does hold in non-zero characteristic.

Lemma 2.10 does not hold in arbitrary characteristic. Let C be the algebraic curve defined by y = 0 in affine coordinates (x, y). Consider the linear system Σ of dimension 1 defined by ϕ_{t}(x, y) := (y = x^{2} + t) in characteristic 2. Then ϕ_{t} is tangent to C at (t^{1/2}, 0) for all t. In particular, (0, 0) is a coincident mobile point for the linear system. The reason for the failure of the lemma is that the function F (x, y) = y −x^{2} defines a Zariski unramified morphism on y = 0 at (0, 0), which is not etale, it is just the Frobenius map in characteristic 2. One can avoid such cases by insisting that the linear system Σ under consideration defines a separable morphism on C (∗). With this extra requirement and a result from [8] (Theorem 6.11) that any locally Zariski unramified separable morphism between curves is locally etale, the proof of Lemma 2.10 holds.

The remaining results of the section are unaffected, with the restriction (∗) on Σ in non-zero characteristic. In particular, Lemma 2.30 holds with this restriction. The proof of the Lemma gives the existence of a generally biunivocal morphism ϕ. By seperability, this induces an isomorphism of the function fields of the respective curves. By an elementary algebraic and model theoretic argument, see for example [2], (Theorem 4.4 p25), one can then invert the morphism ϕ in the sense of Definition 1.19. Therefore, ϕ will define a birational map.

Section 3. The proof of Lemma 3.2 requires results from Section 2 which may not hold in certain exceptional cases. However, the Lemma is still true in arbitrary characteristic. One should replace the use of Lemma 2.30 by invoking general results for plane curves in [6]. Lemma 3.6 and Theorem 3.3 also holds, if we replace birational with biunivocal. In order to obtain the full statement of Theorem 3.3 in arbitrary characteristic, one can use the following argument;

We obtain from the argument of Theorem 3.3, in arbitrary characteristic, a generally biunivocal morphism ϕ from C ⊂ P^{2} to C_{1} ⊂ P^{w}. This induces an inclusion of function fields ϕ^{∗} : L(C_{1}) → L(C). We may factor this extension as L(C_{1}) ⊂ L(D) ⊂ L(C), with D an algebraic curve, L(D) ⊂ L(C) a purely inseperable extension and L(C_{1}) ⊂ L(D) a seperable extension. We, therefore, obtain rational maps ϕ_{1} : C ⇝ D and ϕ_{2} : D ⇝ C_{1}, such that ϕ_{2} ∘ ϕ_{1} is equivalent to ϕ as a biunivocal map between C and C_{1}. Now, using the method of [8] (Remarks 6.5), we may find an algebraic curve C′ ⊂ P^{2} (apply some power of Frobenius to the coefficients defining C) and a morphism F rob^{n} : C → C′, together with a birational map ϕ_{3} : D ↭ C′ such that F rob^{n} and ϕ_{3} ∘ ϕ_{1} are equivalent as biunivocal maps between C and C′. We now obtain a seperable rational map ϕ_{4} = ϕ_{2} ∘ ϕ_{3}^{−1} : C′ ⇝ C_{1}, such that ϕ_{4} ∘ F rob^{n} and ϕ are equivalent as biunivocal morphisms. Now let U ⊂ NonSing(C) be an open set on which ϕ and ϕ_{4} ∘ F rob^{n} are defined and agree as morphisms. By an elementary application of the chain rule and the fact that the differential of the Frobenius morphism is identically zero, one obtains that, for any x ∈ U, (Dϕ)_{x} contains the tangent line l_{x} of C. By the methods in the introduction of Section 1, this is in fact a closed condition on (Dϕ), hence, in fact (Dϕ)_{x} contains the tangent line l_{x} of C, for x ∈ NonSing(C), at any point where ϕ is defined. We can summarise this more generally in the following lemma;

Lemma 7.1. Let ϕ : C ⇝ P^{w} be an inseperable rational map, then, for any nonsingular point x of C at which ϕ is defined, (Dϕ)_{x} contains the tangent line l_{x} of C.

By Remark 3.7, this property is excluded for a transverse g_{n}^{r} as used in Theorem 3.3.

Section 4. The projection construction defined at the beginning of the section may fail to define a separable morphism in non-zero characteristic. However, using Lemma 7.1 and methods from Section 1, one can easily show that this only occurs for projective curves C with the property that, for every x ∈ NonSing(C), the tangent line l_{x} passes through a given point P. In this case, the projection of C from P onto any hyperplane will be inseparable. In [2] such curves are called strange. Non-singular strange curves were completely classified by Samuel in [9];

Theorem 7.2. The only strange non-singular projective algebraic curves are the line and the conic in characteristic 2.

However, there are examples of other singular strange projective algebraic curves in P^{w}, for w ≥ 2, not contained in any hyperplane section. For example, the curves F r_{w} ⊂ P^{w} obtained by iterating Frobenius, given parametrically by;

(t, t^{p}, t^{p^{2}}, …, t^{p^{w−1}}) in characteristic p.

For these examples, Lemma 4.2 fails. In order to see this, pick independent points {T, S} on F r_{w} given by (t, t^{p}, …, t^{p^{w−1}}) and (s, s^{p}, …, s^{p^{w−1}}). Then, the equation of the chord l_{T S} is given parametrically by;

(t + λs, t^{p} + λs^{p}, …, t^{p^{w−1}} + λs^{p^{w−1}})

If t + λs = v, and V is given by (v, v^{p}, …, v^{p^{w−1}}), then we have that the chord l_{T S} meets V, distinct from {T, S}, iff we can solve λ^{p−1} = 1, …, λ^{p^{w−1}−1} = 1 for λ ≠ 1. This is clearly possible if p ≥ 3. In this case, we would have that the chord l_{T S} intersects F r_{w} in at least p points.

Lemma 4.2 holds in arbitrary characteristic, if we exclude singular strange projective curves, however the proof should be modified as it involves Lemma 2.10 applied to a projection. If C is a non-singular strange curve, using the classification given above, the theorem has no content as we assumed that C was not contained in any hyperplane section.

Lemma 7.3. Lemma 4.2 in arbitrary characteristic, excluding singular strange projective curves

Let C ⊂ P^{w}, for w ≥ 3, not contained in any hyperplane section and such that C is not a singular strange projective curve. Suppose that {A, B} are independent generic points of C, then the line l_{AB} does not otherwise meet the curve C.

Proof. We use the same notation as in Lemma 4.2. Let pr_{P} be the projection defined in this lemma. Suppose that pr_{P} is inseperable, then, by the above remarks C is a strange projective algebraic curve, such that all its tangent lines l_{x}, for x ∈ NonSing(C), pass through P. Hence, we can assume that pr_{P} is seperable. We now show that the degenerate case (†) cannot occur. Suppose that pr_{P} (l_{A}) is a point. We have that dim_{P} (A) = 1, hence we can find an open W ⊂ NonSing(C), defined over P, such that, for x ∈ W, l_{x} passes through P. In particular, as dim_{P} (B) = 1, l_{B} passes through P, hence we must have that l_{A} = l_{B} = AB. As A and B were independent, it follows easily that C must be a line l, which is a contradiction. We can now follow through the proof of Lemma 4.2 to obtain that;

There exists an open W ⊂ NonSing(C), defined over the field of definition of C, such that, for y ∈ pr_{B}(W), the l_{y} intersect in a point Q. (∗ ∗ ∗∗)

It follows that, for x ∈ W, the l_{x} intersect l_{B}. In particular, l_{A} intersects l_{B}. If l_{A} = l_{B}, we obtain that C is a line, hence we may assume that l_{A} ∩ l_{B} = Q. As B was generic, we can find an open subset W′ ⊂ NonSing(C), defined over AB, such that, for x ∈ W′, the l_{x} intersect l_{A} and l_{B}. Then, either, for such x ∈ W′, the l_{x} all pass through Q or the l_{x} all lie in the plane P_{AB} defined by l_{A} and l_{B}. In the first case, we have that C is a strange curve, contradicting the hypotheses. In the second case, we use the fact that the plane P l = P_{AB} must be defined over the field of definition of C and then, by the fact that the generic chord l_{AB} lies in P l, that C must be contained in P l as well, contradicting the hypotheses. □

Lemma 4.5 is true if we exclude singular strange projective curves. In order to obtain the corresponding result for a singular strange projective curve C, pick a generic point point P ∈ P^{w}. Let x ∈ C be generic and independent from P. We claim that l_{P x} does not otherwise meet the curve (∗). If not, we can find y ∈ C, distict from x, such that P ∈ l_{xy}. Hence, dim(P/xy) ≤ 1 and dim(P/x) ≥ 3. Now calculate dim(P xy) in two different ways;

(i). dim(P xy) = dim(P/xy) + dim(xy) ≤ 1 + 2 = 3

(ii).dim(P xy) = dim(y/P x) + dim(P x) ≥ 0 + 4 = 4

This clearly gives a contradiction. It follows, using (∗), by an elementary model theoretic argument, that the projection pr_{P} onto any hyperplane H will be generally biunivocal on C. Lemma 4.6 may also be easily modified to include the case of singular strange curves. Theorem 4.8 holds in arbitrary characteristic by the modifications of Lemma 4.5 and Lemma 4.6 and by ensuring that the projections pr_{P} always define seperable morphisms. In the case of strange curves, we can always ensure this by picking the centre of projection P to be disjoint from the bad point Q, defined as the intersection of the tangent lines. Lemma 4.9 is still true in arbitrary characteristic but the proof needs to be modified in order to take into account singular strange projective curves, (we implicitly used Lemma 4.2 in the proof). Using the same notation as in the lemma, given x ∈ C, using the same argument, we can find P ∈ P^{w} generic, such that l_{xP} does not otherwise meet the curve. Now using the modification of Lemma 4.5, the projection from P will be generally biunivocal and, by construction, biunivocal at x. We can then obtain the lemma by repeating this argument. Lemma 4.12 holds in arbitrary characteristic provided the projection pr is seperable. As we have already remarked, this can always be arranged in non-zero characteristic. Lemma 4.14 holds in arbitrary characteristic, using the previous modified lemmas, and the fact that a seperable biunivocal map, between C and pr(C), may be inverted to give a birational map including the nonsingular points of pr(C). It follows that Theorem 4.15 holds in arbitrary characteristic as well, by the modifications from Section 3. Finally, Theorem 4.16 holds by checking the result for certain further unusual curves, depending on generalisations of results in later sections, (see (†) below) . We should note that, without these generalisations, Theorem 4.16 still holds if one accepts the weaker definition of a node as the origin of 2 linear branches (see Definition 6.3).

Section 5. The results of this section hold in arbitrary characteristic up to Lemma 5.24. We only make the remark that it is always possible to choice a maximal linear system such that it defines a separable morphism on a projective curve C. The proof of Lemma 5.24 has the same complications as Lemma 2.10. Again, we can avoid these complications and recover the remaining results of the section in arbitrary characteristic, by the assumption on the linear system Σ that it defines a seperable morphism on C.

Section 6. The results up to Definition 6.3 hold, by appropriate choices of linear systems Σ. In Definition 6.3, the claim that a generic point of an algebraic curve is an ordinary simple point does not hold in arbitrary characteristic, (†). An example is given by the plane quartic curve F (x, y, z) = x^{3}y + y^{3}z + z^{3}x = 0 over a field of characteristic 3. Every point of this curve is an inflection point, that is a point with character (1, 2). In this case, the natural duality map;

DF : C → C^{∗}

[x : y : z] ↦ [F_{x} : F_{y} : F_{z}] = [z^{3} : x^{3} : y^{3}]

is purely inseparable. In order to prove Theorem 4.16 in arbitrary characteristic, one needs to classify such exceptional curves. This can be done, using work of Plucker on the transformation of branches by duality, we save this point of view for another occasion.

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Mathematics Department, The University of Camerino, Camerino, Italy

  E-mail address: tristam.depiro@unicam.it