Brill-Noether theory for curves of a fixed gonality
Abstract
We prove a generalization 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 realizability theorem for tropical stable maps in obstructed geometries, generalizing a well-known theorem of Speyer on genus one curves to arbitrary genus.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2017
- DOI:
- 10.48550/arXiv.1701.06579
- arXiv:
- arXiv:1701.06579
- Bibcode:
- 2017arXiv170106579J
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14H51;
- 14T05
- E-Print:
- 35 pages, 10 TikZ figures. v2: Minor corrections. Final version to appear in Forum of Mathematics, Pi