Multiplicity algebras for rank 2 bundles on curves of small genus

TL;DR

This paper describes the structure of multiplicity algebras associated with fixed points of Higgs bundle moduli spaces for rank 2 bundles on low genus curves, revealing how these algebras vary continuously with the bundle.

Contribution

It provides a detailed description of the multiplicity algebra for specific fixed points, linking algebraic relations to geometric quadrics and analyzing the discriminant's role.

Findings

Relations in the algebra are described by a family of quadrics.

The discriminant varies continuously as the bundle changes.

The algebra's isomorphism class depends smoothly on the bundle.

Abstract

Hausel introduced a commutative algebra -- the multiplicity algebra -- associated to a fixed point of the C^*-action on the Higgs bundle moduli space. Here we describe this algebra for a fixed point consisting of a very stable rank 2 vector bundle and zero Higgs field for a curve of low genus. Geometrically, the relations in the algebra are described by a family of quadrics and we focus on the discriminant of this family, providing a new viewpoint on the moduli space of stable bundles. The discriminant in our examples demonstrates that as the bundle varies, we obtain a continuous variation in the isomorphism class of the algebra.