Multigraded Castelnuovo-Mumford regularity via Klyachko filtrations

TL;DR

This paper develops a geometric approach using Klyachko filtrations to analyze multigraded modules on toric varieties, providing bounds and methods to compute their Hilbert functions and polynomials.

Contribution

It introduces a novel application of Klyachko filtrations to multigraded modules, enabling explicit bounds and computational techniques for Hilbert functions on toric varieties.

Findings

Constructed lattice polytopes encoding multigraded Hilbert functions.

Provided sharp bounds for multigraded regularity indices.

Developed a method to compute Hilbert polynomials for multigraded modules.

Abstract

In this paper, we consider $\mathbb{Z}^{r}-$graded modules on the $\mathrm{Cl}(X)$ $-$graded Cox ring $\mathbb{C}[x_{1},\dotsc,x_{r}]$ of a smooth complete toric variety $X$. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive $\mathbb{Z}^{s+r+2}$-graded modules over non-standard bigraded polynomial rings $\mathbb{C}[x_{0},\dotsc,x_{s},y_{0},\dotsc,y_{r}]$. In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial.