Nontautological Bielliptic Cycles

TL;DR

This paper extends the proof that certain bielliptic curve classes are nontautological, demonstrating this property for genus g ≥ 12 and specifically for genus 12, contributing to the understanding of tautological classes.

Contribution

The authors generalize previous results to show that bielliptic locus classes are nontautological for all genus g ≥ 12, including the case g=12.

Findings

Bielliptic locus classes are nontautological for g ≥ 12.

The result extends previous work from genus 2 to higher genera.

The class for genus 12 is explicitly shown to be nontautological.

Abstract

Let $[\overline{\mathcal{B}}_{2,0,20}]$ and $[\mathcal{B}_{2,0,20}]$ be the classes of the loci of stable resp. smooth bielliptic curves with 20 marked points where the bielliptic involution acts on the marked points as the permutation (1 2)...(19 20). Graber and Pandharipande proved that these classes are nontatoulogical. In this note we show that their result can be extended to prove that $[\overline{\mathcal{B}}_{g}]$ is nontautological for $g\geq 12$ and that $[\mathcal{B}_{12}]$ is nontautological.