English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove a generalization of the Brill-Noether theorem for the variety ofspecial divisors $W^r_d(C)$ on a general curve $C$ of prescribed gonality. Ourmain theorem gives a closed formula for the dimension of $W^r_d(C)$. We buildon previous work of Pflueger, who used an analysis of the tropical divisortheory of special chains of cycles to give upper bounds on the dimensions ofBrill--Noether varieties on such curves. We prove his conjecture, that thisupper bound is achieved for a general curve. Our methods introduce logarithmicstable maps as a systematic tool in Brill-Noether theory. A precise relationbetween the divisor theory on chains of cycles and the corresponding tropicalmaps theory is exploited to prove new regeneration theorems for linear serieswith negative Brill-Noether number. The strategy involves blending an analysisof obstruction theories for logarithmic stable maps with the geometry ofBerkovich curves. To show the utility of these methods, we provide a short newderivation of lifting for special divisors on a chain of cycles with genericedge lengths, proved using different techniques by Cartwright, Jensen, andPayne. A crucial technical result is a new realizability theorem for tropicalstable maps in obstructed geometries, generalizing a well-known theorem ofSpeyer on genus one curves to arbitrary genus.

Brill-Noether theory for curves of a fixed gonality