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Versality of Brill-Noether flags and degeneracy loci of twice-marked curves

Nathan Pflueger

Abstract

A Brill-Noether degeneracy locus is closure in $\Pic^d(C)$ of the locus of line bundles with a specified rank function $r(a,b) = h^0(C,L(-ap-bq))$. These loci generalize the classical Brill-Noether loci $W^r_d(C)$ as well as Brill-Noether loci with imposed ramification. For general $(C,p,q)$ we determine the dimension, singular locus, and intersection class of Brill-Noether degeneracy loci, generalizing classical results about $W^r_d(C)$. The intersection class has a combinatorial interpretation in terms of the number of reduced words for a permutation associated to the rank function, or alternatively the number of saturated chains in the Bruhat order. The essential tool is a versality theorem for a certain pair of flags on $\Pic^d(C)$, conjectured by Melody Chan and the author.

Versality of Brill-Noether flags and degeneracy loci of twice-marked curves

Abstract

A Brill-Noether degeneracy locus is closure in of the locus of line bundles with a specified rank function . These loci generalize the classical Brill-Noether loci as well as Brill-Noether loci with imposed ramification. For general we determine the dimension, singular locus, and intersection class of Brill-Noether degeneracy loci, generalizing classical results about . The intersection class has a combinatorial interpretation in terms of the number of reduced words for a permutation associated to the rank function, or alternatively the number of saturated chains in the Bruhat order. The essential tool is a versality theorem for a certain pair of flags on , conjectured by Melody Chan and the author.

Paper Structure

This paper contains 21 sections, 23 theorems, 45 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The versality theorem
  3. The coupled Petri map
  4. Brill-Noether degeneracy loci
  5. Related work on Brill-Noether theory of marked curves
  6. Conventions
  7. Outline of the paper
  8. Degeneration to chains of twice-marked curves
  9. Valuation of tensors
  10. Adapted bases
  11. Injective coupling
  12. Flags and permutations
  13. Versality of a pair of flags
  14. First-order deformations of a pair of flags
  15. A description of the dual space
  16. ...and 6 more sections

Key Result

Theorem 1.1

If $(C,p,q)$ is a general twice-marked curve of genus $g$, then the pair of degree-$d$ Brill-Noether flags $\mathcal{P}^{{\raisebox{1pt}{\scaleto{\bullet}{2.5pt}}}}_d, \mathcal{Q}^{{\raisebox{1pt}{\scaleto{\bullet}{2.5pt}}}}_d$ of $(C,p,q)$ is versal.

Figures (5)

  • Figure 1: Examples of Brill-Noether degeneracy loci, with $|\Pi| = 3$. The shaded boxes indicate $\mathrm{Ess}(\Pi)$.
  • Figure 2: The arithmetic surface $\mathcal{C} \to \mathrm{Spec} R$ in Situation \ref{['sit:chain']}.
  • Figure 3: The composition $r_{a,b} \circ \delta_x(v)$ as a snake map.
  • Figure 4: Extension of a dot pattern with $r+1 = |\Pi| = 3$ to a $(d,g)$-confined permutation, for various values of $d-g$. The dot pattern in the last example is not $(d,g)$-confined. The dashed squares are explained in Lemma \ref{['lem:bijectiveSquare']}.
  • Figure 5: The number of non-inversions of $\pi$ is $g-\rho_g(d, \Pi)$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 1.2
  • Example 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 2.2
  • Proposition 2.3
  • ...and 39 more