Brill-Noether theory for curves of a fixed gonality

Abstract

We prove a generalisation 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 realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus $1$ curves to arbitrary genus.

(Hurwitz-)Brill-Noether general marked graphs via the Demazure product

“…For the chain of loops with general torsion, lifting results have been obtain in [CJP15] for the case with no marked points, and [He18] in the one-marked point case. For a chain with k-torsion, lifting results for certain splitting loci were obtained in [JR21]. Question 6.4.…”

Recent Developments in Brill-Noether Theory

“…One might reasonably approach such problems starting from [CLRW20] via tropical lifting, as in [CJP15], or via log deformation theory, as in [JR21]. Alternatively, one could look for analogous statements about limit linear series on a double cover of a k-gonal chain of elliptic curves by a folded chain of elliptic curves, and identify which of these are limits of line bundles in the Prym-Brill-Noether locus of a degenerating family of covers, as was done in [LLV20] for the Hurwitz-Brill-Noether theory.…”

Prym–Brill–Noether Loci of Special Curves

“…In this case, the target Γ of the map ϕ is the chain of loops that recently appeared in various celebrated papers (e.g. [JP16,Pfl17a,JR17]). It consists of g loops, denoted by γ 1 , .…”

“…When a curve is not general in moduli, its Brill-Noether locus is no longer expected to be irreducible or pure-dimensional. Nevertheless, the dimensions of irreducible components of these loci have recently been computed for general k-gonal curves, namely general among curves that admit a k-fold cover of P 1 [CPJ19, JR17,Lar19].…”

“…There are numerous interesting avenues for investigation moving forward. The techniques developed in [JR17] for lifting special divisors should be adapted to the current situation to determine the precise dimension of algebraic Prym-Brill-Noether loci (see Conjecture 3.9 for more details). If every maximal cell of V r (Γ, ϕ) can be lifted, Proposition 4.8 would moreover imply that algebraic Prym-Brill-Noether loci are pure-dimensional.…”