Very ampleness in strongly minimal sets

TL;DR

This paper introduces the concept of very ampleness in strongly minimal sets, explores its properties, and characterizes structures in algebraically closed fields that exhibit this property, including implications for definable families and reducts.

Contribution

It defines and studies very ampleness in strongly minimal sets, providing characterizations and applications in algebraically closed fields and groups, and answering longstanding questions.

Findings

Any strongly minimal set internal to an algebraically closed field is very ample.

Very ample strongly minimal sets non-orthogonal to Y are internal to Y.

Divisible strongly minimal groups are very ample.

Abstract

Inspired by very ampleness of Zariski Geometries, we introduce and study the notion of a very ample family of plane curves in any strongly minimal set, and the corresponding notion of a very ample strongly minimal set (characterized by the definability of such a family). We show various basic properties; for example, any strongly minimal set internal to an expansion of an algebraically closed field is very ample, and any very ample strongly minimal set non-orthogonal to a strongly minimal set $Y$ is internal to $Y$. We then apply these results with Zilber's restricted trichotomy to characterize using very ampleness those structures $\mathcal M=(M,\dots)$ interpreted in an algebraically closed field which recover all constructible subsets of powers of $M$. Next we show that very ample strongly minimal sets admit very ample families of plane curves of all dimensions, and use this to characterize very ampleness in terms of definable pseudoplanes. Finally, we show that divisible strongly minimal groups are very ample, and deduce -- answering an old question of G. Martin -- that in a pure algebraically closed field, $K$ there are no reducts between $(K,+,\cdot)$ and $(K, \cdot)$.