Syzygies, multigraded regularity and toric varieties
By Milena Hering, Henry Schenck, Gregory G. Smith. Compositio Mathematica, 2005
58 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/febf6b3a9366b9dc7bb84c11221b6880e24f4faa
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Citations in the corpus
- Projective toric varieties as fine moduli spaces of quiver representations (2006, background): “e the complete quiver of sections for L = (OX,L1,...,Lr). Since Lr is very ample and L1,...,Lr−1 are basepoint-free, it follows that the line bundle L := N i∈Q0 Li is very ample. Since Theorem 2.1 in [HSS] implies that H0(X,L 1) ⊗k ··· ⊗k H0(X,Lr) → H0(X,L) is surjective, Corollary 4.10 implies that”
- Open Problems on Syzygies and Hilbert Functions (2009, background): “The general toric case is considered in [70].”