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 $A^{\ast }(E_N)$ for $N\ge 2$. For each $N\ge 2$, $A^{\ast }(E_N)$ 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 $A^{\ast }(E_2)$ 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.

15 pages, comments welcome!

References in corpus (3)

• Fano threefolds and K3 surfaces
• The intersection theory of the moduli stack of vector bundles on ${\mathbb{P}}^1$
• Arithmetic geometry of the moduli stack of Weierstrass fibrations over ${\mathbb{P}}^1$