Euler characteristics of Brill-Noether varieties

@article{Chan2017EulerCO,
  title={Euler characteristics of Brill-Noether varieties},
  author={Melody Chan and Nathan Pflueger},
  journal={arXiv: Algebraic Geometry},
  year={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 ramification profiles specified at up to two general points. Up to sign, this Euler characteristic is the number of standard set-valued tableaux of a certain skew shape with g labels. We use a flat degeneration via the Eisenbud-Harris theory of limit-linear series, relying on moduli-theoretic advances of Osserman and… 

Figures from this paper

𝐾-classes of Brill–Noether Loci and a Determinantal Formula
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 series
Recent Developments in Brill-Noether Theory
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
A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves
We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d
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 the
Combinatorial relations on skew Schur and skew stable Grothendieck polynomials
We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard
Relative Richardson Varieties
A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base
A pointed Prym-Petri Theorem
We construct pointed Prym-Brill-Noether varieties parametrizing line bundles assigned to an irreducible étale double covering of a curve with a prescribed minimal vanishing at a fixed point. We
Relative Bott-Samelson varieties
We prove that, defined with respect to versal flags, the product of two relative Bott-Samelson varieties over the flag bundle is a resolution of singularities of a relative Richardson variety. This
The Gieseker–Petri theorem and imposed ramification
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 linear
Motivic classes of degeneracy loci and pointed Brill‐Noether varieties
Motivic Chern and Hirzebruch classes are polynomials with K‐theory and homology classes as coefficients, which specialize to Chern–Schwartz–MacPherson classes, K‐theory classes, and Cappell–Shaneson
...
...

References

SHOWING 1-10 OF 43 REFERENCES
𝐾-classes of Brill–Noether Loci and a Determinantal Formula
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 series
Genera of Brill-Noether curves and staircase paths in Young tableaux
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,
Invariants of the Brill–Noether curve
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. When
Limit linear series: Basic theory
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.
Combinatorial relations on skew Schur and skew stable Grothendieck polynomials
We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard
Relative Richardson Varieties
A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base
A geometric Littlewood-Richardson rule
We describe a geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of Schubert varieties. There are no restrictions on the base
Combinatorial Aspects of the K-Theory of Grassmannians
Abstract. In this paper we study Grothendieck polynomials indexed by Grassmannian permutations, which are representatives for the classes corresponding to the structure sheaves of Schubert varieties
Stable Grothendieck polynomials and K-theoretic factor sequences
We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable
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 of
...
...