Cohomology of General Sheaves in Moduli and Existence of Semistable Sheaves on del Pezzo Surfaces
Open Access
Author:
Levine, Daniel
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
February 28, 2020
Committee Members:
Jack William Huizenga, Dissertation Advisor/Co-Advisor Jack William Huizenga, Committee Chair/Co-Chair Yuri G Zarkhin, Committee Member Ping Xu, Committee Member Martin Bojowald, Outside Member Carina Pamela Curto, Program Head/Chair
Keywords:
Algebraic Geometry Sheaf Cohomology Moduli of Sheaves del Pezzo Surfaces Rational Surfaces Moduli Spaces
Abstract:
Let $X_m$ be a del Pezzo surface of degree $9-m$, and let $L \in \Pic(X_m)$ be the total transform of a line on $\PP^2$. When $m \leq 5$, we compute the cohomology of a general sheaf in $M(\bv)$, the moduli space of Gieseker semistable sheaves with Chern character $\bv$. Let $W_m$ be the Weyl group acting on $\Pic(X_m)$. More generally, we compute the cohomology of a general sheaf in a larger irreducible algebraic stack $\cP_{L^W}(\bv)$, the stack of torsion-free sheaves satisfying $\Ext^2(\cE,\cE(-\sigma(L))) = 0$ for all $\sigma \in W_m$. From this computation, we are able to classify the Chern characters for which the general sheaf in $M(\bv)$ is non-special, i.e. has at most one nonzero cohomology group. When $m \leq 6$, we show our construction of certain direct sums of line bundles implies the existence of stable and semistable sheaves with respect to the anti-canonical polarization. This dissertation is based on work in \cite{LZBrillNoether}.
