On the local behaviour of symmetric differentials on the blow-up of Du Val singularities

Abstract: Du Val singularities appear in the classification of algebraic surfaces and other areas of algebraic geometry. Wahl's concept of local Euler characteristics of sheaves helps in describing the properties of these singularities. We consider the sheaf of symmetric differentials and compute one ingredient of the local Euler characteristic: the codimension of those symmetric differentials that extend to the blow-up of the singularity in the space of those that are regular around it. For singularities of type \(A_n\), we show that this codimension can be expressed combinatorially as a lattice point count in a polytope. Ehrhart's quasi-polynomials allow us to find closed expressions for this codimension as a function of the symmetric differential degree. We expect our method to generalize to all Du Val singularities.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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