Geodesic
Brill-Noether theory for curves of a fixed gonality
Forum of Mathematics, Pi, Tome 9 (2021)

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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. 
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David Jensen; Dhruv Ranganathan. Brill-Noether theory for curves of a fixed gonality. Forum of Mathematics, Pi, Tome 9 (2021). doi: 10.1017/fmp.2020.14

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