On A Tautological Relation Conjectured By Buryak-Shadrin

Abstract

Buryak and Shadrin conjectured a tautological relation on moduli spaces of curves ${\overline{\mathcal{M}}}_{g,n}$ which has the form $B_{g,\text{d}}^m=0$ for certain tautological classes $B_{g,\text{d}}^m$ where $m\ge 2,n\ge 1$ and $|\text{d}|\ge 2g+m-1$. In this paper we prove that this conjecture holds if it is true for the $m=2$ and $|\text{d}|=2g+1$ case. This result reduces the proof of this conjecture to checking finitely many cases for each genus $g$. We will also prove this conjecture for the $g=1$ case.