Tautological rings of stable map spaces

TL;DR

This paper provides a complete presentation of the tautological rings of genus zero stable map spaces, offering generators and relations with a clear geometric interpretation, especially for target projective spaces.

Contribution

It introduces a set of generators and relations for extended tautological rings of genus zero stable map spaces, with explicit descriptions for projective space targets.

Findings

Complete presentation of cohomology and Chow rings for genus zero stable maps to projective spaces.

Generators and relations are geometrically interpreted.

Results apply to cases with and without marked points.

Abstract

We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is $\PP^n$, these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points.