Brill-Noether theory for curves of a fixed gonality

• Published in: Forum of mathematics. Pi Vol. 9
• Main Authors: Jensen, David, Ranganathan, Dhruv
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 2021
• Subjects: 14H51; 14T05; Algebraic and Complex Geometry; Chains; Curves; Regeneration; Theorems; Upper bounds; 14T05; 14H51
• ISSN: 2050-5086; 2050-5086
• DOI: 10.1017/fmp.2020.14

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.