Tropical methods in Hurwitz-Brill-Noether theory

Kaelin Cook-Powell, David Jensen

Research output: Contribution to journalArticlepeer-review

Abstract

Splitting type loci are the natural generalizations of Brill-Noether varieties for curves with a distinguished map to the projective line. We give a tropical proof of a theorem of H. Larson, showing that splitting type loci have the expected dimension for general elements of the Hurwitz space. Our proof uses an explicit description of splitting type loci on a certain family of tropical curves. We further show that these tropical splitting type loci are connected in codimension one, and describe an algorithm for computing their cardinality when they are zero-dimensional. We provide a conjecture for the numerical class of splitting type loci, which we confirm in a number of cases.

Original languageEnglish
Article number108199
JournalAdvances in Mathematics
Volume398
DOIs
StatePublished - Mar 26 2022

Bibliographical note

Funding Information:
We would like to thank Melody Chan, Hannah Larson, and Yoav Len, who each provided helpful insights during their visits to the University of Kentucky in 2019–2020. We thank Yoav Len, Sam Payne, and Dhruv Ranganathan for helpful comments on an earlier draft of this paper, and Gavril Farkas for telling us about his paper [14] , which was the inspiration for Example 7.12 . We thank Eric Larson, Hannah Larson, and Isabel Vogt for pointing us toward the existing literature on k-cores and the affine symmetric group. This work was supported by NSF DMS-1601896 .

Funding Information:
We would like to thank Melody Chan, Hannah Larson, and Yoav Len, who each provided helpful insights during their visits to the University of Kentucky in 2019?2020. We thank Yoav Len, Sam Payne, and Dhruv Ranganathan for helpful comments on an earlier draft of this paper, and Gavril Farkas for telling us about his paper [14], which was the inspiration for Example 7.12. We thank Eric Larson, Hannah Larson, and Isabel Vogt for pointing us toward the existing literature on k-cores and the affine symmetric group. This work was supported by NSF DMS-1601896.

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

  • Brill-Noether theory
  • Gonality
  • Hurwitz spaces
  • Tropical geometry

ASJC Scopus subject areas

  • Mathematics (all)

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