Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-15T21:02:22.499Z Has data issue: false hasContentIssue false

Syzygies, multigraded regularity and toric varieties

Published online by Cambridge University Press:  24 November 2006

Milena Hering
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USAmhering@umich.edu
Hal Schenck
Affiliation:
Mathematics Department, Texas A&M University, College Station, TX 77843, USAschenck@math.tamu.edu
Gregory G. Smith
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, Canadaggsmith@mast.queensu.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety $X$. Given globally generated line bundles $B_{1}, \dotsc, B_{\ell}$ on $X$ and $m_{1}, \dotsc, m_{\ell} \in \mathbb{N}$, consider the line bundle $L := B_{1}^{m_{1}} \otimes \dotsb \otimes B_{\ell}^{m_{\ell}}$. We give conditions on the $m_{i}$ which guarantee that the ideal of $X$ in $\mathbb{P}(H^{0}(X,L)^{*})$ is generated by quadrics and that the first $p$ syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2006