Submission date: 8 March 2019

Abstract:

A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system (for k ≥ 2) is a linear space such that each line has size exactly k. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (bi-interpretable) vocabulary τ with a single ternary relation R. We prove that for every integer k there exist 2^{ℵ_0}-many integer valued functions μ such that each μ determines a distinct strongly minimal Steiner k-system G_μ, whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.

Mathematics Subject Classification: 03C45, 51E10

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Full text arXiv 1903.03541: pdf, ps.