Abstract:

סמינר סיום מאסטר של אלן סורני

מנחים ד"ר הווארד נואר ופרופח דני נפטין

The Bogomolov-Gieseker Inequality for Surfaces with Rational Double Points כותרת

אבסטרקט

The Bogomolov-Gieseker inequality for smooth projective surfaces in characteristic zero is the inequality $\mathrm{ch}_1\left({\mathcal{E}\right)^2 - 2 \mathrm{ch}_0\left(\mathcal{E}\right) . \mathrm{ch}_2\left(\mathcal{E}\right) \geq 0$, where $\mathcal{E}$ is a slope semistable coherent sheaf. The inequality has many applications including the boundedness of the moduli space of semistable sheaves, explicit restriction theorems for stable sheaves, and the existence of Bridgeland stability conditions which have been applied to solve many classical problems in algebraic geometry. In this talk we give an algebraic proof of the Bogomolov-Gieseker inequality on projective surfaces with at worst rational double point singularities. Such singular points are of particular interest as they are the possible canonical singularities of surfaces, i.e. those singularities arising in running the minimal model program on surfaces.