A local cohomology obstruction to small Cohen--Macaulay modules

Source abstract

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.