The integral Chow ring of $\mathcal{M}_{0}(\mathbb{P}^r, d)$, for $d$ odd

TL;DR

This paper provides a presentation of the integral Chow ring of the moduli stack of degree d maps from genus zero curves to projective space for odd d, including generators, relations, and explicit examples.

Contribution

It introduces a new presentation for the Chow ring of al_{0}(\u211d^r, d) for odd d, with an efficient set of generators and relations, and offers explicit computations and conjectures.

Findings

Presented a polynomial ring quotient description of the Chow ring.

Derived an efficient set of generators for the ideal of relations.

Computed explicit examples for low values of d and r.

Abstract

For any odd integer $d$, we give a presentation for the integral Chow ring of the stack $\Mcal_{0}(\Pro^r, d)$, as a quotient of the polynomial ring $\Z[c_1,c_2]$. We describe an efficient set of generators for the ideal of relations, and compute them in generating series form. The paper concludes with explicit computations of some examples for low values of $d$ and $r$, and a conjecture for a minimal set of generators.