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Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

Forum Mathematicum , Volume 34 (1): 20 – Jan 1, 2022

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Publisher
de Gruyter
Copyright
© 2022 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2021-0143
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Forum Mathematicumde 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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