On the Picard Group of the Moduli Space of Curves via $r$-Spin Structures

TL;DR

This paper proves Pixton's conjecture for the second rational cohomology of the moduli space of curves by deriving explicit relations and comparing them with established theorems, advancing understanding of the Picard group structure.

Contribution

The paper provides explicit expressions for relations in the cohomology of moduli spaces and confirms Pixton's conjecture in this context, linking $r$-spin structures with cohomological relations.

Findings

Confirmed Pixton's conjecture for the second cohomology

Derived explicit relations for the Picard group

Connected $r$-spin structures with cohomological relations

Abstract

In this paper, we obtain explicit expressions for Pandharipande-Pixton-Zvonkine relations in the second rational cohomology of $\overline{\mathcal{M}}_{g,n}$ and comparing the result with Arbarello-Cornalba's theorem we prove Pixton's conjecture in this case.