The integral Chow ring of $\mathscr{M}_{0}(\mathbb{P}^r, 2)$

TL;DR

This paper computes the integral Chow rings of moduli stacks of degree 2 maps from rational curves to projective space, revealing a rich combinatorial structure in the relations.

Contribution

It provides a presentation of these Chow rings as polynomial quotients, with relations encoded by rational generating functions.

Findings

Chow rings are presented as quotients of polynomial rings.

Relations are encoded by two rational generating functions.

The structure varies with the dimension r of projective space.

Abstract

We compute a presentation for the integral Chow rings of the moduli stacks of degree $2$ maps from smooth rational curves to projective space $\mathbb{P}^r$, as a quotient of a three-variable polynomial ring. The relations as $r$ varies have rich combinatorial structure: all non-trivial relations are encoded by two generating functions which are rational functions.