The Chow rings of moduli spaces of elliptic surfaces over ℙ¹

2021/10/10 by Samir Canning, Canning, Samir, Bochao Kong +1
Mathematics · #14C15 #14C17 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2110.04928

openalex publication_date 2021/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E_N denote the coarse moduli space of smooth elliptic surfaces over ℙ¹ with fundamental invariant N. We compute the Chow ring A^*(E_N) for N≥ 2. For each N≥ 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^*(E₂) are tautological classes on the moduli space 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.

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