# Genera of Brill-Noether curves and staircase paths in Young tableaux

@article{Chan2015GeneraOB,
title={Genera of Brill-Noether curves and staircase paths in Young tableaux},
author={Melody Chan and Alberto L'opez Mart'in and Nathan Pflueger and Montserrat Teixidor I. Bigas},
journal={arXiv: Algebraic Geometry},
year={2015}
}
• M. Chan, +1 author M. Bigas
• Published 2015
• Mathematics
• arXiv: Algebraic Geometry
In this paper, we compute the genus of the variety of linear series of rank $r$ and degree $d$ on a general curve of genus $g$, with ramification at least $\alpha$ and $\beta$ at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves… Expand

#### Figures from this paper

Euler characteristics of Brill-Noether varieties
• Mathematics
• 2017
We prove an enumerative formula for the algebraic Euler characteristic of Brill-Noether varieties, parametrizing degree d and rank r linear series on a general genus g curve, with ramificationExpand
𝐾-classes of Brill–Noether Loci and a Determinantal Formula
• Mathematics
• 2017
We prove a determinantal formula for the K-theory class of certain degeneracy loci, and apply it to compute the Euler characteristic of the structure sheaf of the Brill-Noether locus of linear seriesExpand
Connectedness of Brill-Noether loci via degenerations
We show that limit linear series spaces for chains of curves are reduced. Using new advances in the foundations of limit linear series, we then use degenerations to study the question ofExpand
Brill-Noether loci with ramification at two points
We prove the injectivity of the Petri map for linear series on a general curve with given ramification at two generic points. We also describe the components of such a set of linear series on a chainExpand
Monodromy and K-theory of Schubert curves via generalized jeu de taquin
• Mathematics
• 2015
We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves$$S(\lambda _\bullet )$$S(λ∙), which are one-dimensional Schubert problems defined withExpand
Versality of Brill-Noether flags and degeneracy loci of twice-marked curves
A line bundle on a smooth curve C with two marked points determines a rank function r(a, b) = h(C,L(−ap − bq)). This paper studies Brill-Noether degeneracy loci; such a locus is defined to be theExpand
The Gieseker–Petri theorem and imposed ramification
• Mathematics
• Bulletin of the London Mathematical Society
• 2019
We prove a smoothness result for spaces of linear series with prescribed ramification on twice-marked elliptic curves. In characteristic 0, we then apply the Eisenbud-Harris theory of limit linearExpand
Schubert curves in the orthogonal Grassmannian.
• Mathematics
• 2019
We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(\lambda_\bullet)$ in the orthogonal Grassmannian $\mathrm{OG}_n$, which are one-dimensional Schubert problemsExpand
On Order Ideals of Minuscule Posets III: The CDE Property
Recent work of Hopkins establishes that the lattice of order ideals of a minuscule poset satisfies the coincidental down-degree expectations property of Reiner, Tenner, and Yong. His approach appealsExpand
On the $q$-Enumeration of Barely Set-Valued Tableaux and Plane Partitions
• Mathematics
• 2021
Barely set-valued tableaux are a variant of Young tableaux in which one box contains two numbers as its entry. It has recently been discovered that there are product formulas enumerating certainExpand

#### References

SHOWING 1-10 OF 37 REFERENCES
The Brill–Noether curve and Prym-Tyurin varieties
We prove that the Jacobian of a general curve C of genus $$g=2a+1$$, with $$a\ge 2$$, can be realized as a Prym-Tyurin variety for the Brill–Noether curve $$W^{1}_{a+2}(C)$$. As consequence of thisExpand
Invariants of the Brill–Noether curve
• Mathematics
• 2014
Abstract For a projective nonsingular curve of genus g, the Brill–Noether locus Wdr(C)$W^r_d(C)$ parametrizes line bundles of degree d over C with at least r + 1 (linearly independent) sections. WhenExpand
Brill–Noether loci in codimension two
Abstract Let us consider the locus in the moduli space of curves of genus $2k$ defined by curves with a pencil of degree $k$. Since the Brill–Noether number is equal to $- 2$, such a locus hasExpand
On the connectedness of degeneracy loci and special divisors
• Mathematics
• 1981
Introduction Let C be a smooth complex projective curve of genus g, and let J be the Jacobian of C. Upon choosing a base-point in C, J may be identified with the set of linear equivalence classes ofExpand
On linear series with negative Brill-Noether number
Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basic object of study is $\mathcal{W}^r_{d,g}$, the moduli space of smooth genus $g$ curves with aExpand
Pointed Castelnuovo numbers
• Mathematics
• 2015
The classical Castelnuovo numbers count linear series of minimal degree and fixed dimension on a general curve, in the case when this number is finite. For pencils, that is, linear series ofExpand
Limit linear series moduli stacks in higher rank
In order to prove new existence results in Brill-Noether theory for rank-2 vector bundles with fixed special determinant, we develop foundational definitions and results for limit linear series ofExpand
A limit linear series moduli scheme
We develop a new, more functorial construction for the basic theory of limit linear series, which provides a compactification of the Eisenbud-Harris theory, and shows promise for generalization toExpand
A simple characteristic-free proof of the Brill-Noether theorem
FollowingWelters, we describe howthe use of a different degeneration from that considered by Eisenbud and Harris leads to a simple and characteristic-independent proof of the Brill-Noether theoremExpand
Limit linear series: Basic theory
• Mathematics
• 1986
AbstractIn this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves, curves of compact type.Expand