Submission date: 5 November 2012.

Abstract:

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety, generic hyperplane sections are geometrically irreducible and codimension one. We also study hypersurface sections in families. In the case of families of hyperplanes, we also establish smoothness results. Following the main theorem, several applications of this work relating to the definability of Kolchin polynomials and the definability of irreducibility in families of differential algebraic varieties are given.

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Full text arXiv 1211.0972: pdf, ps.