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Syzygies, multigraded regularity and toric varieties

Published online by Cambridge University Press:  24 November 2006

Milena Hering
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109,
Hal Schenck
Mathematics Department, Texas A&M University, College Station, TX 77843,
Gregory G. Smith
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6,
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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.

Research Article
Foundation Compositio Mathematica 2006