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AbstractIn this paper, we consider ℤr{\mathbb{Z}^{r}}-graded modules on the Cl(X){\operatorname{Cl}(X)}-graded Cox ring ℂ[x1,…,xr]{\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 ℤs+r+2{\mathbb{Z}^{s+r+2}}-graded modules over any non-standard bigraded polynomial ring ℂ[x0,…,xs,y0,…,yr]\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.
Forum Mathematicum – de Gruyter
Published: Jan 1, 2022
Keywords: Klyachko filtrations; Castelnuovo–Mumford regularity; Cox rings; Hilbert function; Hilbert polynomial; 14M25; 13D40; 14F06; 13A02
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