Cactus varieties of sufficiently ample embeddings of projective schemes have determinantal equations

TL;DR

This paper proves that cactus varieties of sufficiently ample embeddings of projective schemes are set-theoretically defined by minors of matrices with linear entries, extending known results and relating to longstanding conjectures.

Contribution

It establishes that cactus varieties of sufficiently ample embeddings are defined by minors, generalizing previous results and connecting to Eisenbud-Koh-Stillman conjectures.

Findings

Cactus varieties are set-theoretically defined by minors of linear matrices.

Sufficiently ample line bundles ensure projective normality.

The work relates to and extends conjectures on determinantal equations for embeddings.

Abstract

For a fixed projective scheme X, a property P of line bundles is satisfied by sufficiently ample line bundles if there exists a line bundle L_0 on X such that P(L) holds for any L with (L - L_0) ample. As an example, sufficiently ample line bundles are very ample, moreover, for a normal variety X, the embedding corresponding to sufficiently ample line bundle is projectively normal. The grandfather of such properties and a basic ingredient used to study this concept is Fujita vanishing theorem, which is a strengthening of Serre vanishing to sufficiently ample line bundles. The r-th cactus variety of X is an analogue of secant variety and it is defined using linear spans of finite schemes of degree r. In this article we show that cactus varieties of sufficiently ample embeddings of X are set-theoretically defined by minors of matrices with linear entries. The topic is closely related to conjectures of Eisenbud-Koh-Stillman, which was proved by Ginensky in the case X a smooth curve. On the other hand Sidman-Smith proved that the ideal of sufficiently ample embedding of any projective scheme X is generated by 2 x 2 minors of a matrix with linear entries.