The Bielliptic Locus in Genus 11

Abstract

The Chow ring of ${\mathcal{M}}_{\mathit{g}}$ is known to be generated by tautological classes for $\mathit{g}\le 9$. Meanwhile, the first example of a nontautological class on ${\mathcal{M}}_{\mathit{g}}$ is the fundamental class of the bielliptic locus in ${\mathcal{M}}_{12}$, due to van Zelm. It remains open if the Chow rings of ${\mathcal{M}}_{10}$ and ${\mathcal{M}}_{11}$ are generated by tautological classes. In these cases, a natural first place to look is at the bielliptic locus. In genus 10, it is already known that classes supported on the bielliptic locus are tautological. Here, we prove that all classes supported on the bielliptic locus are tautological in genus 11. By Looijenga’s vanishing theorem, this implies that they all vanish.