The bielliptic locus in genus 11

TL;DR

This paper proves that all classes supported on the bielliptic locus in genus 11 are tautological, extending known results from genus 10 and contributing to understanding the structure of the Chow ring of moduli spaces.

Contribution

It establishes that in genus 11, classes supported on the bielliptic locus are tautological, a new result that advances knowledge of the Chow ring structure.

Findings

All classes supported on the bielliptic locus in genus 11 are tautological.

By Looijenga's vanishing theorem, these classes vanish in genus 11.

Extends previous results from genus 10 to genus 11.

Abstract

The Chow ring of $\mathcal{M}_g$ is known to be generated by tautological classes for $g \leq 9$. Meanwhile, the first example of a non-tautological class on $\mathcal{M}_{g}$ 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.