Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes

TL;DR

This paper investigates when secant varieties of Veronese embeddings are defined by determinantal equations, providing positive results for low dimensions and counterexamples for higher dimensions linked to Gorenstein schemes.

Contribution

It establishes conditions under which secant varieties are set-theoretically cut out by catalecticant minors, and relates failure cases to smoothability of Gorenstein schemes.

Findings

Secant varieties are defined by minors of catalecticant matrices for dimensions ≤ 3.

Counterexamples show catalecticant minors are insufficient in higher dimensions.

Connection between secant varieties and smoothability of Gorenstein schemes.

Abstract

We study the secant varieties of the Veronese varieties and of Veronese reembeddings of a smooth projective variety. We give some conditions, under which these secant varieties are set-theoretically cut out by determinantal equations. More precisely, they are given by minors of a catalecticant matrix. These conditions include the case when the dimension of the projective variety is at most 3 and the degree of reembedding is sufficiently high. This gives a positive answer to a set-theoretic version of a question of Eisenbud in dimension at most 3. For dimension four and higher we produce plenty of examples when the catalecticant minors are not enough to set-theoretically define the secant varieties to high degree Veronese varieties. This is done by relating the problem to smoothability of certain zero-dimensional Gorenstein schemes.