Double total ramifications for curves of genus 2

Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P^1 of degree d totally ramified over the two marked points, for d>= 2. Such loci have codimension two. We compute the class of their compactifications in the moduli space of stable curves. Several results will be deduced from this computation.Comment: Final version to appear in International Mathematics Research Notice

Du Val curves and the pointed Brill-Noether Theorem

We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.Comment: 15 pages. Final version, to appear in Selecta Mathematic

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 with special vanishing at marked points. When the Brill-Noether number $\rho$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $\rho=1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan-L\'opez-Pflueger-Teixidor for the arithmetic genus of a Brill-Noether curve of special divisors. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations, and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey-Jockusch-Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for skew tableaux.Comment: 31 pages; v2: stronger Theorem C, and improved expositio

Brill-Noether loci in codimension two

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 has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.Comment: v2: new section 7, the main Theorem is now proved for ALL k>=3. Final version to appear in Compositio Mathematica. This version is slightly differen