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
Summary
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].
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