Finite schemes and secant varieties over arbitrary characteristic

TL;DR

This paper develops scheme theoretic methods to study secant varieties over arbitrary fields, focusing on finite schemes and their smoothability, with results applicable across different characteristics and field extensions.

Contribution

It generalizes secant variety theory to arbitrary characteristic and base fields, emphasizing the independence of smoothability from embeddings and field extensions.

Findings

Secant varieties behave well under high degree Veronese reembeddings.

Secant varieties are defined by minors of catalecticants under smoothability conditions.

Smoothability of finite schemes is independent of embeddings and base field extensions.

Abstract

We present scheme theoretic methods that apply to the study of secant varieties. This mainly concerns finite schemes and their smoothability. The theory generalises to the base fields of any characteristic, and even to non-algebraically closed fields. In particular, the smoothability of finite schemes does not depend on the embedding into a smooth variety or on base field extensions. Independent of the base field, secant varieties to high degree Veronese reembeddings behave well with respect to the intersection and they are defined by minors of catalecticants whenever a suitable smoothability condition for Gorenstein subschemes holds. The content of the article is largely expository, although many results are presented in a stronger form than in the literature.