BrillNoether theory for curves of a fixed gonality
Abstract
We prove a generalization of the BrillNoether 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 BrillNoether 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 BrillNoether 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 BrillNoether 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 realizability theorem for tropical stable maps in obstructed geometries, generalizing a wellknown theorem of Speyer on genus one curves to arbitrary genus.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 DOI:
 10.48550/arXiv.1701.06579
 arXiv:
 arXiv:1701.06579
 Bibcode:
 2017arXiv170106579J
 Keywords:

 Mathematics  Algebraic Geometry;
 14H51;
 14T05
 EPrint:
 35 pages, 10 TikZ figures. v2: Minor corrections. Final version to appear in Forum of Mathematics, Pi