Abstract

AbstractWe prove a generalisation of the Brill-Noether theorem for the variety of special divisors$W^r_d(C)$on a general curveCof 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.

Highlights

  • We prove a generalisation of the Brill-Noether theorem for the variety of special divisors ( ) on a general curve C of prescribed gonality

  • We focus on the case of chains of cycles in this text, but the procedural aspects of our proof should generalise to new combinatorial geometries and give high genus realisability theorems

  • If we consider the image of in P + −1 under the linear series in | | described above, any points whose sum is in the class of span a projective space of dimension − 1

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Summary

Introduction

Given a smooth projective curve over the complex numbers, let ( ) denote the subvariety of the Picard variety of , paramterising divisors of degree that move in a linear system of dimension at least. The dimensions of these varieties are fundamental invariants of. The dimension of ( ) is ( , , ) := − ( + 1) ( − + ), where a scheme is understood to be empty when its dimension is negative This result was first proved in a seminal paper by Griffiths and Harris [29].

Context and Motivation
Combinatorics of Special Chains of Cycles
Torsion Profiles and Displacement Tableaux
Coordinates on Γ and Its Picard Group
Lingering Lattice Paths
Vertex Avoiding Divisors and Bases of Rational Functions
Deformations of Maps
Logarithmic Prestable Maps
Tropical Realisability
Lifting Divisors on a Generic Chain of Cycles
Extended Example
Maps to Scrolls
Scrollar Tableaux
A Lifting Result for Maps to Toric Varieties
Strategy
Statement of the Lifting Criterion
Maps to the Projective Line
Maps to Toric Targets
Lifting Divisors on a -Gonal Chain of Cycles
Full Text
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