Relations in the Tautological Ring of the Universal Curve

TL;DR

This paper establishes bounds on the dimensions of the tautological ring of the universal curve for various genera, leading to a complete description for genus up to 9 and confirming its Gorenstein property.

Contribution

It provides new bounds and exact structures for the tautological ring of the universal curve, especially for genus up to 9, and demonstrates its Gorenstein property in these cases.

Findings

Tautological ring is Gorenstein for genus up to 9.

Bounds on dimensions are established for genus up to 27.

Exact structure of the tautological ring is determined for genus up to 9.

Abstract

We bound the dimensions of the graded pieces of the tautological ring of the universal curve from below for genus up to 27 and from above for genus up to 9. As a consequence we obtain the precise structure of the tautological ring of the universal curve for genus up to 9. In particular, we see that it is Gorenstein for these genera.