A rank-2 vector bundle on \mathbb P²× \mathbb P² and projective geometry of nonclassical Enriques surfaces in characteristic 2

2025/04/22 by Ziv Ran, Ran, Ziv, Jürgen Rathmann +1
Mathematics · Medicine · #14j60 #14n05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Magnolia and Illicium research

paper · pdf · doi:10.48550/arxiv.2504.16174

openalex publication_date 2025/04/22 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

We construct a rank-2 indecomposable vector bundle on \mathbb P²×\mathbb P² in characteristic 2 that does not come from a bundle on \mathbb P² by factor projection nor from a bundle on \mathbb P^m by central projection. We show that the zero-sets of a suitable twist of E form a family of nonclassical smooth Enriques surfaces of bidegree (4, 4) whose general member is 'singular' in the sense that Frobenius acts isomorphically on H¹, and there is a smooth divisor consisting of smooth supersingular surfaces (Frobenius acts as zero). Every nonclassical Enriques surface of bidegree (4, 4) in \mathbb P²×\mathbb P² that is bilinearly normal arises as a zero-set in this way.

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