发表机构
School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出局部上同调障碍，证明在特定曲面完备截环上，当数值条件c(Y,H)>0时，不存在非零有限极大Cohen--Macaulay模，并给出六线Hirzebruch--Kummer族实例。
AI 中文摘要
我们提出一个局部上同调障碍，用于曲面完备截环上小Cohen--Macaulay模的存在性。设Y为光滑连通复射影曲面，H为丰沛整体生成除子，且数值上K_Y≡4H。令c(Y,H)=15H^2/8-χ(O_Y)。主要断言是：当c(Y,H)>0时，环⊕_{n≥0}H^0(Y,O_Y(nH))的完备顶点局部环上的每个非零有限自反模M满足dim_C H^2_m(M)≥c(Y,H)rk M。论证结合了自反扩张的初等变换、Harder--Narasimhan斜率、Bogomolov不等式以及二截面Koszul估计。一个固定源核估计将所得上同调转移到穿孔谱，而不需要M上的分次。对于六线Hirzebruch--Kummer族的六次成员，H^2=1080且χ(O_Y)=1926，因此界为99 rk M。我们给出该应用的显式完全交模型，并详细说明导致所声称的非零有限极大Cohen--Macaulay模不存在性的论证。
英文摘要
We propose a local-cohomology obstruction to the existence of small Cohen--Macaulay modules over completed section rings of surfaces. Let $Y$ be a smooth connected complex projective surface, and let $H$ be an ample globally generated divisor with $K_Y\equiv4H$ numerically. Put $c(Y,H)=15H^2/8-χ(\OO_Y)$. The main claim is that, when $c(Y,H)>0$, every nonzero finite reflexive module $M$ over the completed vertex local ring of $\bigoplus_{n\geq0}H^0(Y,\OO_Y(nH))$ satisfies $\dim_\C H^2_\mm(M)\geq c(Y,H)\rk M$. The argument combines elementary transformations of reflexive extensions, Harder--Narasimhan slopes, Bogomolov's inequality, and a two-section Koszul estimate. A fixed-source kernel estimate transfers the resulting cohomology to the punctured spectrum without requiring a grading on $M$. For the degree-six member of the six-line Hirzebruch--Kummer family, $H^2=1080$ and $χ(\OO_Y)=1926$, so the bound is $99\rk M$. We give an explicit complete-intersection model for this application and spell out the argument leading to the claimed nonexistence of nonzero finite maximal Cohen--Macaulay modules.
Comments20 pages
