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Transmission permutations and Demazure products in Hurwitz--Brill--Noether theory

Nathan Pflueger

Abstract

A line bundle on a curve with two marked points can be special in many ways, as measured by the global sections of all of its twists by these points. All of this information is conveniently packaged into a permutation, which we call the transmission permutation. We prove that when twice-marked curves are chained together, these permutations are composed via the Demazure product; in reverse, bundles with given permutation can be enumerated via reduced decompositions of a permutation. This paper demonstrates the utility of transmission permutations by giving a short derivation of the basic dimension bounds of both classical Brill--Noether theory and Hurwitz--Brill--Noether theory in a unified framework. The difference between the two cases derives from taking permutations in either symmetric groups or affine symmetric groups.

Transmission permutations and Demazure products in Hurwitz--Brill--Noether theory

Abstract

A line bundle on a curve with two marked points can be special in many ways, as measured by the global sections of all of its twists by these points. All of this information is conveniently packaged into a permutation, which we call the transmission permutation. We prove that when twice-marked curves are chained together, these permutations are composed via the Demazure product; in reverse, bundles with given permutation can be enumerated via reduced decompositions of a permutation. This paper demonstrates the utility of transmission permutations by giving a short derivation of the basic dimension bounds of both classical Brill--Noether theory and Hurwitz--Brill--Noether theory in a unified framework. The difference between the two cases derives from taking permutations in either symmetric groups or affine symmetric groups.

Paper Structure

This paper contains 18 sections, 20 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The transmission permutation
  3. Transmission loci
  4. The Demazure product
  5. Results
  6. The Demazure product on ASP
  7. Transmission permutations on smooth curves and chains
  8. Chains of twice-marked curves
  9. The genus 1 case
  10. Relative transmission loci
  11. Versal deformation of a chain
  12. Relative transmission loci
  13. Brill--Noether varieties and splitting loci
  14. The essential set
  15. Brill--Noether loci
  16. ...and 3 more sections

Key Result

Theorem 1.7

A very general twice-marked curve $(C,p,q)$ in $\mathcal{M}_{g,2}$ has $0$-general transmission, and such curves are Brill--Noether general; for all $k \geq 2$ a very general point in some component of $\mathcal{H}_{g,k,2}$ has $k$-general transmission, and such curves are Hurwitz--Brill--Noether ge

Figures (2)

  • Figure 1: Two examples of transmission permutations, and what they say about $h^0$ and $h^1$ of twists $\mathcal{L}(ap-bq)$ of $\mathcal{L}$. See Examples \ref{['eg:iota']} and \ref{['eg:bitangent']}.
  • Figure 2: A versal deformation of a chain, and the divisors $Y_i, Y'_i$.

Theorems & Definitions (49)

  • Remark 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 2.1: pflDemazure
  • Definition 2.2
  • Theorem 2.3: pflDemazure
  • ...and 39 more