# The tautological ring of $\mathcal{M}_{g,n}$ via Pandharipande-Pixton-Zvonkine $r$-spin relations

## TL;DR

This paper employs relations from the Givental formula for $r$-spin classes to derive new bounds and proofs regarding the dimensions of tautological rings of moduli spaces of curves, providing alternative proofs for known results.

## Contribution

It introduces new proofs and bounds for the dimensions of tautological rings of $arm_{g,n}$ using relations from the $r$-spin Givental formula, enhancing understanding of their structure.

## Key findings

- Proof that $ ext{dim } R^{g-1}(arm_{g,n}) ext{ } extless extgreater n$
- Proof that $R^{i}(arm_{g,n})=0$ for $i extgreater extgreater g$
- Estimates for $ ext{dim } R^{i}(arm_{g,n})$ for $i extless extless g-2$

## Abstract

We use relations in the tautological ring of the moduli spaces $\overline{\mathcal{M}}_{g,n}$ derived by Pandharipande, Pixton, and Zvonkine from the Givental formula for the $r$-spin Witten class in order to obtain some restrictions on the dimensions of the tautological rings of the open moduli spaces $\mathcal{M}_{g,n}$. In particular, we give a new proof for the result of Looijenga (for $n=1$) and Buryak et al. (for $n\geq 2$) that $\dim R^{g-1}(\mathcal{M}_{g,n}) \leq n$. We also give a new proof of the result of Looijenga (for $n=1$) and Ionel (for arbitrary $n\geq 1$) that $R^{i}(\mathcal{M}_{g,n}) =0$ for $i\geq g$ and give some estimates for the dimension of $R^{i}(\mathcal{M}_{g,n})$ for $i\leq g-2$.