射影簇上的上同调消失、Koszul上同调与多重分次正则性
Cohomology Vanishing, Koszul Cohomology and Multigraded Regularity on Projective Varieties
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中文总结 AI 辅助
本文在多重分次Castelnuovo–Mumford正则性框架下，证明射影簇的通用上同调消失定理，推广前期工作，刻画射影空间乘积线丛的极小正则性，重证Green定理并计算超椭圆曲线典范像的Betti表。
中文摘要 AI 辅助
本文在多重分次Castelnuovo–Mumford正则性框架下，证明了任意射影簇上的一个通用上同调消失定理。特别地，将该消失定理应用于证明Koszul上同调群$K_{p,q}(X;F,L)$的消失，其中$L=B_1^{w_1}\boxtimes\boldsymbol{\times}B_t^{w_t}$，线丛$B_1,\boldsymbol{\times},B_t$全局生成，$F$为向量丛。还引入$K_{p,q}$层级，为性质$(N_p)$和$(M_q)$的消失准则提供统一视角，同时阐明混合权合冲的消失，这些结果推广了文献\textbf{[HeringSchenckSmith]}、\textbf{[GallegoPurnaprajnaII]}与\textbf{[Basu]}的前期工作。此外，完全刻画了任意射影空间乘积上的线丛的极小多重分次正则性，通过直接正则性论证重新得到射影空间的Green消失定理，给出另一证明。最后，结合所证消失定理与Green对偶，计算了超椭圆曲线典范像的完全分次Betti表。
英文摘要
In this article, we prove a general cohomology vanishing theorem on arbitrary projective varieties within the framework of multigraded Castelnuovo--Mumford regularity. In particular, we apply this vaishing theorem to prove the vanishing of Koszul cohomology groups $K_{p,q}(X;F,L)$, where $L=B_1^{w_1}\otimes\cdots\otimes B_t^{w_t}$, the line bundles $B_1,\ldots,B_t$ are globally generated, and $F$ is a vector bundle. We also introduce the $K_{p,q}$-hierarchy, providing a unified perspective on the vanishing criteria for Properties $(N_{p})$ and $(M_{q})$ while shedding light on the vanishing of mixed-weight syzygies. These results generalize earlier work in \cite{HeringSchenckSmith}, \cite{GallegoPurnaprajnaII}, and \cite{Basu}. Furthermore, we give a complete description of minimal multigraded regularities of line bundles on arbitrary products of projective spaces. We also recover Green's vanishing theorem for projective space via a direct regularity argument, giving an alternative proof. Finally, we compute the complete graded Betti table of the canonical image of a hyperelliptic curve by combining our vanishing theorem with Green's duality.
