Obstructions to unirationality for product-quotient surfaces over \mathbbF_p

2025/08/20 by Church, Benjamin
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric and Algebraic Topology #Primary: 14E08. Secondary: 14G17 and 14J29

paper · pdf · doi:10.48550/arxiv.2508.14876

openalex publication_date 2025/08/20 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

We construct a surface over \mathbbF_p with π₁^ét(X) = 1 that is supersingular -- in the sense that H²_ét(X, ℚ_ℓ(1)) is spanned by algebraic cycles -- but is not unirational. This provides a counterexample to a 1977 conjecture of Shioda. To achieve this, we produce new obstructions to unirationality for product-quotient surfaces.

Citations

• Local Euler characteristics of A_n-singularities and their application to hyperbolicity
• Local vanishing and Hodge filtration for rational singularities
• Algebraic curves uniformized by congruence subgroups of triangle groups
• Explicit methods for Hilbert modular forms
• The classification of minimal product-quotient surfaces with p_g=0
• Computing systems of Hecke eigenvalues associated to Hilbert modular forms
• Quotients of products of curves, new surfaces with p_g=0 and their fundamental groups
• Heights of Heegner points on Shimura curves

Related

• Product-Quotient Surfaces: Result and Problems
• On the unirationality of supersingular K3 surfaces
• Unirationality of certain supersingular K3 surfaces in characteristic 5
• Product-Quotient Surfaces: new invariants and algorithms
• Supersingular K3 surfaces in odd characteristic and sextic double planes