Cactus Varieties of Sufficiently Ample Embeddings of Projective Schemes Have Determinantal Equations

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