The Chow rings of moduli spaces of elliptic surfaces over $\mathbb{P}^1$
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 $10N2$. 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.04928
 Bibcode:
 2021arXiv211004928C
 Keywords:

 Mathematics  Algebraic Geometry;
 14C15;
 14C17
 EPrint:
 15 pages, comments welcome!