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Brill-Noether loci and strata of differentials

Andrei Bud

Abstract

We prove that the projectivized strata of differentials are not contained in pointed Brill-Noether divisors, with only a few exceptions. For a generic element in a stratum of differentials, we show that many of the associated pointed Brill-Noether loci are of expected dimension. We use our results to study the Auel-Haburcak Conjecture: We obtain new non-containments between maximal Brill-Noether loci in $\mathcal{M}_g$. Our results regarding quadratic differentials imply that the quadratic strata in genus $6$ are uniruled.

Brill-Noether loci and strata of differentials

Abstract

We prove that the projectivized strata of differentials are not contained in pointed Brill-Noether divisors, with only a few exceptions. For a generic element in a stratum of differentials, we show that many of the associated pointed Brill-Noether loci are of expected dimension. We use our results to study the Auel-Haburcak Conjecture: We obtain new non-containments between maximal Brill-Noether loci in . Our results regarding quadratic differentials imply that the quadratic strata in genus are uniruled.
Paper Structure (10 sections, 20 theorems, 102 equations)

This paper contains 10 sections, 20 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Intersection of Brill-Noether divisors with projectivized strata of differentials
  3. Maximal Brill-Noether loci and the Auel-Haburcak Conjecture
  4. Birational geometry
  5. Prym curves and strata of quadratic differentials
  6. Strata of differentials and Brill-Noether divisors
  7. Positive pointed Brill-Noether number
  8. Maximal Brill-Noether loci
  9. Uniruledness of quadratic strata in genus $6$
  10. Quadratic strata and Prym-Brill-Noether loci

Key Result

Theorem 1.1

Let $r\geq 1$, $g\geq r+2$ be natural numbers and $a = (0\leq a_0<a_1<\cdots a_r \leq g+r)$ a vanishing sequence satisfying and $a_r\leq g$. Then, if $[C,p]$ is generic in the stratum $\mathcal{H}_g^{\textrm{odd}}(2g-2)$, it does not admit a $g^r_{g+r}$ with prescribed vanishing $a$ at $p$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • ...and 22 more