The integral Chow rings of moduli of Weierstrass fibrations

TL;DR

This paper computes the integral Chow rings of moduli stacks of minimal Weierstrass fibrations over the projective line, providing explicit generators and relations, especially for rational and K3 elliptic surfaces.

Contribution

It provides a quotient stack presentation and a complete description of the Chow rings for all minimal Weierstrass fibrations, including explicit cases for rational and K3 surfaces.

Findings

Chow rings are explicitly computed with generators and relations.

Provides quotient stack presentations for moduli stacks.

Special cases for rational and K3 elliptic surfaces are detailed.

Abstract

We compute the Chow rings with integral coefficients of moduli stacks of minimal Weierstrass fibrations over the projective line. For each integer $N\geq 1$, there is a moduli stack $\mathcal{W}^{\mathrm{min}}_N$ parametrizing minimal Weierstrass fibrations with fundamental invariant $N$. Following work of Miranda and Park--Schmitt, we give a quotient stack presentation for each $\mathcal{W}^{\mathrm{min}}_N$. Using these presentations and equivariant intersection theory, we determine a complete set of generators and relations for each of the Chow rings. For the cases $N=1$ (respectively, $N=2$), parametrizing rational (respectively, K3) elliptic surfaces, we give a more explicit computation of the relations.