Gotzmann's persistence theorem for smooth projective toric varieties

TL;DR

This paper generalizes Gotzmann's persistence theorem from projective space to smooth projective toric varieties, showing that the Hilbert polynomial can be confirmed by checking the Hilbert function at a number of points depending only on the Picard rank.

Contribution

The authors extend Gotzmann's persistence theorem to smooth projective toric varieties, with the number of points depending solely on Picard rank, independent of dimension.

Findings

Generalization of Gotzmann's theorem to toric varieties

Number of points to check depends only on Picard rank

Applicable to all smooth projective toric varieties

Abstract

Gotzmann's persistence theorem enables us to confirm the Hilbert polynomial of a subscheme of projective space by checking the Hilbert function in just two points, regardless of the dimension of the ambient space. We generalise this result to products of projective spaces, and then extend our result to any smooth projective toric variety. The number of points to check depends solely on the Picard rank of the ambient space, with no dependence on the dimension.