TL;DR
This paper computes the class of the closure of a specific Brill-Noether locus in the moduli space of genus 2k curves with a pencil of degree k, using test surfaces, revealing its codimension two structure.
Contribution
It introduces a method to explicitly compute the class of the Brill-Noether locus in the moduli space of stable curves for genus 2k.
Findings
Determined the class of the Brill-Noether locus closure
Confirmed the locus has codimension two
Applied test surface method successfully
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.
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