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Prym-Brill-Noether theory for ramified double covers

Andrei Bud

Abstract

We initiate the study of Prym-Brill-Noether theory for ramified double covers, extending several key results from classical Prym-Brill-Noether theory to this new framework. In particular, we improve Kanev's results on the dimension of pointed Prym-Brill-Noether loci for ramified double covers. Additionally, we compute the dimension of twisted Prym-Brill-Noether loci with vanishing conditions at points, thus extending the results of Tarasca. Furthermore, we compute the class of the twisted Prym-Brill-Noether loci inside (a translation of) the Prym variety, thus extending the results of de Concini and Pragacz to ramified double covers. Finally, we prove that a generic Du Val curve is Prym-Brill-Noether general.

Prym-Brill-Noether theory for ramified double covers

Abstract

We initiate the study of Prym-Brill-Noether theory for ramified double covers, extending several key results from classical Prym-Brill-Noether theory to this new framework. In particular, we improve Kanev's results on the dimension of pointed Prym-Brill-Noether loci for ramified double covers. Additionally, we compute the dimension of twisted Prym-Brill-Noether loci with vanishing conditions at points, thus extending the results of Tarasca. Furthermore, we compute the class of the twisted Prym-Brill-Noether loci inside (a translation of) the Prym variety, thus extending the results of de Concini and Pragacz to ramified double covers. Finally, we prove that a generic Du Val curve is Prym-Brill-Noether general.

Paper Structure

This paper contains 13 sections, 28 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Moduli spaces of ramified curves and Brill-Noether conditions
  3. Prym-Brill-Noether loci for ramified double covers
  4. Twisted Prym-Brill-Noether loci
  5. Boundary degenerations and Prym-Brill-Noether general curves
  6. Introduction to Prym-Brill-Noether Theory
  7. Prym-Brill-Noether loci as Lagrangian degenerations
  8. Brill-Noether and Gieseker-Petri conditions
  9. Prym-Brill-Noether Theory for ramified double covers
  10. Twisted Prym-Brill-Noether loci
  11. Prym-Brill-Noether loci twisted by an effective divisor
  12. Prym-Brill-Noether and pointed Brill-Noether conditions
  13. Du Val curves and Prym-Brill-Noether generality

Key Result

Theorem 1.1

Let $f\colon \widetilde{C}\rightarrow C$ be a generic double cover in $\mathcal{R}_{g,2}$ or $\mathcal{R}_{g,4}$. Then the curve $\widetilde{C}$ is Gieseker-Petri general.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • proof
  • ...and 41 more