A refined Brill–Noether theory over Hurwitz spaces

Larson, Hannah

doi:10.1007/s00222-020-01023-z

Cited by 15 publications

(17 citation statements)

References 15 publications

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“…Indeed, this conjectural description has been confirmed in the years since this paper first appeared. The Brill-Noether variety ( ) admits a stratification by isomorphism classes of scrolls, and each stratum has a certain expected dimension [38,20].…”

Section: Approach and Techniquesmentioning

confidence: 99%

Brill-Noether theory for curves of a fixed gonality

Jensen

Ranganathan

2021

Forum of Mathematics, Pi

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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.

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Section: Approach and Techniquesmentioning

confidence: 99%

Brill-Noether theory for curves of a fixed gonality

Jensen

Ranganathan

2021

Forum of Mathematics, Pi

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“…This analog of Theorem 1.2(1) has been followed by more comprehensive results, giving a description of all components of W r d (C), along with their dimensions and basic properties. The starting point for this subsequent work is a stratification of W r d (C), using the following discrete invariants that generalize the Maroni invariants of trigonal curves, introduced independently by H. Larson [Lar21] and by Cook-Powell and the first author [CJ19].…”

Section: Theorem 23 ([Jr21]mentioning

confidence: 99%

“…A general trigonal curve is not hyperelliptic, so W 2 4 (C) is empty and thus, as discussed in Example 2.2, W 1 4 (C) has two irreducible components, one of dimension one and the other an isolated point of dimension 0. These results are proven by degeneration to a chain of elliptic curves in [Lar21,LLV20]. To ensure that this degenerate curve admits a map of degree k to a rational curve, one specifies that the difference between the two attaching points on each component has torsion order k in the group law on the elliptic curve.…”

Section: Theorem 23 ([Jr21]mentioning

confidence: 99%

Recent Developments in Brill-Noether Theory

Jensen¹,

Payne²

2021

Preprint

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We briefly survey recent results related to linear series on curves that are general in various moduli spaces, highlighting the interplay between algebraic geometry on a general curve and the combinatorics of its degenerations. Breakthroughs include the proof of the Maximal Rank Theorem, which determines the Hilbert function of the general linear series of given degree and rank on the general curve in Mg, and complete analogs of the standard Brill-Noether theorems for curves that are general in Hurwitz spaces. Other advances include partial results in a similar direction for linear series in the Prym locus of a general unramified double cover of a general k-gonal curve and instances of the Strong Maximal Rank Conjecture.

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“…Enumerative questions have been of particular interest recently, including examining (as in this paper) many intersection products and structure constants on M g,n [3,26], tautological relations [7,21,22], and Schubert calculus involving limit linear series [5,11]. Other topics of interest include the S n action on H * (M 0,n ) over C [1,14,24] and R [23], Chern classes of vector bundles on M 0,n associated to sl r [9,15], explicit projective equations for M 0,n [20], and similar questions pertaining to a number of closely-related moduli spaces [6,8,13,19,25].…”

Section: Introductionmentioning

confidence: 99%

Lazy tournaments and multidegrees of a projective embedding of $\overline{M}_{0,n}$

Gillespie¹,

T.²,

Levinson³

2021

Preprint

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We provide a new geometric interpretation of the multidegrees of the (iterated) Kapranov embedding Φn : M 0,n+3 → P 1 × P 2 × • • • × P n , where M 0,n+3 is the moduli space of stable genus 0 curves with n + 3 marked points. We enumerate the multidegrees by disjoint sets of boundary points of M 0,n+3 via a combinatorial algorithm on trivalent trees that we call a lazy tournament. These sets are compatible with the forgetting maps used to derive the recursion for the multidegrees proven in 2020 by Gillespie, Cavalieri, and Monin.The lazy tournament points are easily seen to total (2n − 1)!! = (2n − 1)giving a natural proof of the fact that the total degree of Φn is the odd double factorial. This fact was first proven using an insertion algorithm on certain parking functions, and we additionally give a bijection to those parking functions.

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A refined Brill–Noether theory over Hurwitz spaces

Cited by 15 publications

References 15 publications

Brill-Noether theory for curves of a fixed gonality

Brill-Noether theory for curves of a fixed gonality

Recent Developments in Brill-Noether Theory

Lazy tournaments and multidegrees of a projective embedding of $\overline{M}_{0,n}$

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