Multigraded Castelnuovo-Mumford regularity via Klyachko filtrations

Abstract

In this paper, we consider ${\mathbb{Z}}^r-$graded modules on the $\mathrm{Cl}(X)$ $-$graded Cox ring $\mathbb{C}[x_1,\dots ,x_r]$ 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 $\mathbb{C}[x_0,\dots ,x_s,y_0,\dots ,y_r]$. 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.

To appear in Forum Mathematicum