A remark on the rational cohomology of $\bar{S}_{1,n}$

TL;DR

This paper investigates the rational cohomology of the moduli space of genus 1 spin curves with marked points, establishing vanishing results for certain cohomology groups and describing the structure of the second cohomology.

Contribution

It proves the vanishing of the first and third rational cohomology groups and shows the second is generated by boundary classes, providing new insights into the topology of this moduli space.

Findings

H^1 and H^3 of the space vanish

H^2 is generated by boundary classes

Provides structural understanding of the cohomology groups

Abstract

We focus on the rational cohomology of Cornalba's moduli space of spin curves of genus 1 with $n$ marked points. In particular, we show that both its first and its third cohomology group vanish and the second cohomology group is generated by boundary classes.