The Class of the Bielliptic Locus in Genus 3

Abstract

Let the bielliptic locus be the closure in the moduli space of stable curves of the locus of smooth curves that are double covers of genus 1 curves. In this paper, we compute the class of the bielliptic locus in the moduli space \overline{M}_3 of stable curves of genus three in terms of a standard basis of the rational Chow group of codimension-2 classes in the moduli space. Our method is to test the class on the hyperelliptic locus: this gives the desired result up to two free parameters, which are then determined by intersecting the locus with two surfaces in \overline{M}_3 .