Gotzmann's persistence theorem for Mori dream spaces

TL;DR

This paper extends Gotzmann's persistence theorem from projective spaces and toric varieties to Mori dream spaces, providing new tools for understanding their Hilbert functions and polynomials.

Contribution

It generalizes the analogue of Gotzmann's theorem to Mori dream spaces with finitely generated Cox rings, and offers a stronger persistence result for products of projective spaces.

Findings

Established an analogue of Gotzmann's theorem for Mori dream spaces.

Provided an alternative, stronger persistence result for points in products of projective spaces.

Extended the applicability of Hilbert polynomial determination methods.

Abstract

Gotzmann's persistence theorem provides a method for determining the Hilbert polynomial of a subscheme of projective space by evaluating the Hilbert function at only two points, irrespective of the dimension of the ambient space. In arXiv:2405.02275 we established an analogue of Gotzmann's persistence theorem for smooth projective toric varieties. We generalise our results to the setting of Mori dream spaces, whose associated Cox rings are also finitely generated. We also give an alternative, stronger, persistence result for points in products of projective spaces.