论文标题
多面规律性的界限
Bounds on Multigraded Regularity
论文作者
论文摘要
多式的castelnuovo-模块$ m $的定期性超过总坐标环$ s $ s $ s $ s $ s $ $ x $是一个区域$ \ operatatorName {reg} m \ subset \ subset \ subset \ subset \ operatotorname {pic} $ nef cone $ \ peragatation x $ variant x $ nef cone $ \ operatatration。我们证明，有限生成的忠实模块的多条规律性包含在$ \ operatatorName {nef} x $的翻译中，由$ m $的发电机的学位确定，因此仅包含有限的最低元素。我们表明，如果$ m $具有扭转，并且PICARD组的排名至少为两个，那么这种情况即使对于循环模块也可能失败。 作为一种应用，我们表现出渐近界限的理想能力。对于$ s $中的$ i $，我们通过证明它包含$ \ operatotorName {reg} s $的翻译，并包含在$ \ operatatorNorname {nef} x $的翻译中，每个界限由固定的vectory a $ n $ n $ n $ n $ n bys $ n transce，我们限制了$ s $中的理想。
Multigraded Castelnuovo--Mumford regularity of a module $M$ over the total coordinate ring $S$ of a smooth projective toric variety $X$ is a region $\operatorname{reg} M \subset \operatorname{Pic} X$ invariant under translation by the nef cone $\operatorname{Nef} X$. We prove that the multigraded regularity of a finitely generated faithful module is contained in a translate of $\operatorname{Nef} X$ determined by the degrees of the generators of $M$, and thus contains only finitely many minimal elements. We show that this condition can fail even for cyclic modules if $M$ has torsion and the rank of the Picard group is at least two. As an application, we exhibit asymptotic bounds for the multigraded regularity of powers of ideals. For $I$ an ideal in $S$, we bound $\operatorname{reg}(I^n)$ by proving that it contains a translate of $\operatorname{reg} S$ and is contained in a translate of $\operatorname{Nef} X$, where each bound translates by a fixed vector as $n$ increases.
