Syzygies, multigraded regularity and toric varieties

[摘要] 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.

[学科分类] 数学（综合）