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Brill-Noether theory for curves of a fixed gonality

Part of: Curves

Published online by Cambridge University Press:  08 January 2021

David Jensen
Affiliation:
Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY40506, USA; E-mail: dave.jensen@uky.edu.
Dhruv Ranganathan
Affiliation:
Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY40506, USA; E-mail: dave.jensen@uky.edu. Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, University of Cambridge, CambridgeCB2 1TP, UK; E-mail: dr508@cam.ac.uk.

Abstract

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We prove a generalisation of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$ . We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus $1$ curves to arbitrary genus.

Type
Algebraic and Complex Geometry
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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