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Towards Brill-Noether Theory for Spectral Curves

Clemens Nollau

TL;DR

This work develops Brill-Noether theory for spectral curves via the Beauville–Narasimhan–Ramanan correspondence, linking line bundles on spectral curves to Higgs bundles on the base. It analyzes three spectral-curve contexts—classical Hitchin curves, spectral covers over $P^1$, and canonical double covers—by employing restricted Hitchin maps, relative section spaces, and endomorphism bounds to obtain dimensions and smoothness of splitting loci and Brill-Noether loci. It determines the gonality and gonality sequences of canonical covers in general and several special cases, showing how Brill-Noether data lifts along covers in low-rank regimes. The paper also provides explicit equations and computational examples for canonical covers, illustrating the practical tractability of the Higgs-bundle perspective in concrete settings.

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.

Towards Brill-Noether Theory for Spectral Curves

TL;DR

This work develops Brill-Noether theory for spectral curves via the Beauville–Narasimhan–Ramanan correspondence, linking line bundles on spectral curves to Higgs bundles on the base. It analyzes three spectral-curve contexts—classical Hitchin curves, spectral covers over , and canonical double covers—by employing restricted Hitchin maps, relative section spaces, and endomorphism bounds to obtain dimensions and smoothness of splitting loci and Brill-Noether loci. It determines the gonality and gonality sequences of canonical covers in general and several special cases, showing how Brill-Noether data lifts along covers in low-rank regimes. The paper also provides explicit equations and computational examples for canonical covers, illustrating the practical tractability of the Higgs-bundle perspective in concrete settings.

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.
Paper Structure (13 sections, 33 theorems, 162 equations, 1 figure)

This paper contains 13 sections, 33 theorems, 162 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on Spectral Curves
  3. Relative section space
  4. Spectral curves with a rational base
  5. The restricted Hitchin map
  6. Quadrics of rank at most $3$ containing a curve
  7. The restricted Hitchin map of the trivial vector bundle of rank $2$
  8. The twisted endomorphisms of an extension
  9. Application to vector bundles with two sections
  10. Pencils on classical spectral curves
  11. The Gonality of canonical covers
  12. The rest of the Gonality Sequence
  13. Equations for specific Canonical Covers

Key Result

Theorem 1.1

HlarsonLLV Let $\psi: C \to \mathbb{P}^1$ be a general cover of degree $d$ and genus $g$. Then $U^{(e,m)}(C)$ has dimension if $\rho(g,e,m)$ is non-negative. Otherwise the splitting locus is empty. Furthermore $U^{(e,m)}(C)$ is irreducible if $\rho(g,e,m) > 0$.

Figures (1)

  • Figure 1: A line bundle $L \in W^1_d(\widetilde{C})$ gives rise to a Higgs bundle $(V,\phi)$. The diagram shows the various possibilities for the vector bundle $V$.

Theorems & Definitions (86)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 76 more