Towards Brill-Noether Theory for Spectral Curves

TL;DR

This paper explores Brill-Noether loci of various spectral curves using the Beauville-Narasimhan-Ramanan correspondence, analyzing their dimensions, smoothness, and gonality sequences in different geometric contexts.

Contribution

It extends Brill-Noether theory to spectral curves of different types, providing detailed dimension counts, smoothness conditions, and gonality results under general assumptions.

Findings

Dimensions of splitting loci computed

Conditions for smoothness established

Gonality sequences determined for low-rank cases

Abstract

We study Brill-Noether loci of three kinds of spectral curves: classical spectral curves as introduced by Hitchin, spectral curves over the projective line and double covers whose branch locus is a canonical divisor. Our techniques are based on the Beauville-Narasimhan-Ramanan correspondence: We push down line bundles on the spectral curve to the base curve and then we study the Higgs bundles obtained in this way. For the first kind we study the spaces of pencils in the Picard variety of a classical spectral curve in detail. In the case of spectral curves over the projective line we deal with their splitting loci which refine the Brill-Noether loci in the Picard variety. We compute their dimensions and investigate whether they are smooth. For the third kind we determine the gonality sequence when the rank of the linear system is much smaller than the genus. For this the base curve and the branch divisor are assumed to be general.