Complete descriptions of the tautological rings of the moduli spaces of curves of genus lower or equal to 4 with marked points

TL;DR

This paper provides comprehensive descriptions of the tautological rings for moduli spaces of genus 1 to 4 curves with marked points, detailing their structure across all degrees.

Contribution

It offers the first complete characterization of the tautological rings for these low-genus moduli spaces, including explicit group descriptions in all degrees.

Findings

Complete descriptions of tautological rings for genus 1-4

Explicit group structures in all degrees

Foundational for understanding moduli space geometry

Abstract

We study here the tautological rings of the moduli spaces of compact Riemann surfaces of genus 1,2,3 and 4 with marked points. The paper presents the complete descriptions of these rings by describing the groups of all degrees.