Syzygies, multigraded regularity and toric varieties

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.

利用多重分次的卡斯泰尔诺沃 - 芒福德正则性，我们研究定义一个簇\(X\)的射影嵌入的方程。给定\(X\)上整体生成的线丛\(B_1,\cdots,B_{\ell}\)以及\(m_1,\cdots,m_{\ell}\in\mathbb{N}\)，考虑线丛\(L := B_1^{m_1}\otimes\cdots\otimes B_{\ell}^{m_{\ell}}\)。我们给出关于\(m_i\)的条件，这些条件保证\(X\)在\(\mathbb{P}(H^{0}(X,L)^*)\)中的理想由二次型生成，并且前\(p\)个合冲是线性的。这在环面簇的合冲以及多面体的正规性方面产生了新的结果。