Tropical methods in Hurwitz-Brill-Noether theory

Kaelin Cook-Powell; David Jensen

doi:10.1016/j.aim.2022.108199

Tropical methods in Hurwitz-Brill-Noether theory

Kaelin Cook-Powell, David Jensen

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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 language	English
Article number	108199
Journal	Advances in Mathematics
Volume	398
DOIs	https://doi.org/10.1016/j.aim.2022.108199
State	Published - Mar 26 2022

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Funding

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 . 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.

Funders	Funder number
National Science Foundation (NSF)	DMS-1601896
University of Kentucky

Keywords

Brill-Noether theory
Gonality
Hurwitz spaces
Tropical geometry

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2022.108199

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AB - 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.

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Tropical methods in Hurwitz-Brill-Noether theory

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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