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 has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.

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 different