Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

Abstract

Abstract In this paper, we consider ℤ r {\mathbb{Z}^{r}} -graded modules on the Cl ⁡ ( X ) {\operatorname{Cl}(X)} -graded Cox ring ℂ ⁢ [ x 1 , … , x r ] {\mathbb{C}[x_{1},\ldots,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 ℤ s + r + 2 {\mathbb{Z}^{s+r+2}} -graded modules over any non-standard bigraded polynomial ring ℂ ⁢ [ x 0 , … , x s , y 0 , … , y r ] \mathbb{C}[x_{0},\ldots,x_{s},\allowbreak y_{0},\ldots,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.