Gotzmann's persistence theorem for Mori dream spaces

TL;DR

This work extends Gotzmann's persistence framework beyond projective space to Mori dream spaces by exploiting the finitely generated Cox ring and toric embeddings, showing that a finite set of evaluation points suffices to determine multigraded Hilbert polynomials. It establishes the existence of Hilbert polynomials for homogeneous ideals on Cox rings and proves a Gotzmann-style persistence theorem with at most $2^s$ evaluation points, where $s$ is the Picard rank, by transferring toric results to the Mori dream space setting. For zero-dimensional subschemes in products of projective spaces, the paper provides a stronger persistence result via Gasharov bounds, enabling verification with a small, explicit set of corner points (as illustrated in a detailed example). Collectively, the results broaden the applicability of persistence techniques to a broad class of varieties with finitely generated Cox rings, offering practical finite-verification criteria for Hilbert polynomials in geometric modeling and computational algebraic geometry.

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.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Theorem 2.6
• proof
• proof : Proof of Theorem \ref{['theorem1']}