The Chow rings of moduli spaces of elliptic surfaces over $\mathbb{P}^1$

TL;DR

This paper computes the Chow rings of moduli spaces of elliptic surfaces over P^1, revealing their Gorenstein structure and applications to subvariety dimensions, with special focus on K3 surfaces polarized by a hyperbolic lattice.

Contribution

It provides the first explicit computation of the Chow rings for these moduli spaces and links the generators to tautological classes, supporting conjectures on K3 surface moduli.

Findings

Chow rings are Gorenstein with socle in codimension 16.

Maximal dimension of complete subvarieties is 16.

Generators for the Chow ring of elliptic K3 surfaces are tautological classes.

Abstract

Let $E_N$ denote the coarse moduli space of smooth elliptic surfaces over $\mathbb{P}^1$ with fundamental invariant $N$. We compute the Chow ring $A^*(E_N)$ for $N\geq 2$. For each $N\geq 2$, $A^*(E_N)$ is Gorenstein with socle in codimension $16$, which is surprising in light of the fact that the dimension of $E_N$ is $10N-2$. As an application, we show that the maximal dimension of a complete subvariety of $E_N$ is $16$. When $N=2$, the corresponding elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice $U$. We show that the generators for $A^*(E_2)$ are tautological classes on the moduli space $\mathcal{F}_{U}$ of $U$-polarized K3 surfaces, which provides evidence for a conjecture of Oprea and Pandharipande on the tautological rings of moduli spaces of lattice polarized K3 surfaces.