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Maximal Brill-Noether loci via degenerations and double covers

Andrei Bud, Richard Haburcak

TL;DR

The paper develops a degenerative approach to Brill-Noether theory using limit linear series on chains and double covers. It shows that closures of certain BN loci contain products of pointed BN loci of small codimension, enabling dimensionally sharp non-containments between maximal BN loci and a new proof of the existence of components of expected dimension in a broad range. It also extends non-containment results through Prym theory by examining images of unramified double covers under the source-curve map, producing infinite families of new non-containments. Overall, the work clarifies the containment lattice among BN loci and provides robust degeneration-based tools for refined BN questions, including Prym-related cases.

Abstract

Using limit linear series on chains of curves, we show that closures of certain Brill--Noether loci contain a product of pointed Brill--Noether loci of small codimension. As a result, we obtain new non-containments of Brill--Noether loci, in particular that all dimensionally expected non-containments hold for expected maximal Brill--Noether loci. Using these degenerations, we also give a new proof that Brill--Noether loci with expected codimension $-ρ\leq \lceil g/2\rceil$ have a component of the expected dimension. Additionally, we obtain new non-containments of Brill--Noether loci by considering the locus of the source curves of unramified double covers.

Maximal Brill-Noether loci via degenerations and double covers

TL;DR

The paper develops a degenerative approach to Brill-Noether theory using limit linear series on chains and double covers. It shows that closures of certain BN loci contain products of pointed BN loci of small codimension, enabling dimensionally sharp non-containments between maximal BN loci and a new proof of the existence of components of expected dimension in a broad range. It also extends non-containment results through Prym theory by examining images of unramified double covers under the source-curve map, producing infinite families of new non-containments. Overall, the work clarifies the containment lattice among BN loci and provides robust degeneration-based tools for refined BN questions, including Prym-related cases.

Abstract

Using limit linear series on chains of curves, we show that closures of certain Brill--Noether loci contain a product of pointed Brill--Noether loci of small codimension. As a result, we obtain new non-containments of Brill--Noether loci, in particular that all dimensionally expected non-containments hold for expected maximal Brill--Noether loci. Using these degenerations, we also give a new proof that Brill--Noether loci with expected codimension have a component of the expected dimension. Additionally, we obtain new non-containments of Brill--Noether loci by considering the locus of the source curves of unramified double covers.
Paper Structure (9 sections, 20 theorems, 68 equations)

This paper contains 9 sections, 20 theorems, 68 equations.

Table of Contents

  1. Background
  2. Brill--Noether loci
  3. Expected maximal Brill--Noether loci
  4. Limit linear series and pointed Brill--Noether loci
  5. Prym--Brill--Noether loci
  6. Non-containments of pointed Brill--Noether loci of small codimension
  7. Dimensionally expected non-containments
  8. Existence of components of expected dimension
  9. Non-containments obtained from Prym

Key Result

Theorem 1

Let $\mathcal{M}^{r}_{g,d}$ and $\mathcal{M}^{s}_{g,e}$ be expected maximal Brill--Noether loci. If $\rho(g,s,e)<\rho(g,r,d)$, then $\mathcal{M}^{r}_{g,d} \nsubseteq \mathcal{M}^{s}_{g,e}$.

Theorems & Definitions (36)

  • Conjecture 1: Maximal Brill--Noether Loci Conjecture
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1.1: ahl_BN_via_gonality_2023
  • Lemma 1.2
  • proof
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • ...and 26 more