A variety that cannot be dominated by one that lifts

2019/02/21 by Remy van Dobben de Bruyn, de Bruyn, Remy van Dobben
Computer Science · Mathematics · #11G25 (Secondary) #14D06 #14F35 #14G17 (Primary) 14D15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1902.07885

openalex publication_date 2019/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a precise version of a theorem of Siu and Beauville on morphisms to higher genus curves, and use it to show that if a variety X in characteristic p lifts to characteristic 0, then any morphism X → C to a curve of genus g ≥ 2 can be lifted along. We use this to construct, for every prime p, a smooth projective surface X over \mathbb F_p that cannot be rationally dominated by a smooth proper variety Y that lifts to characteristic 0.

Citations

• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I
• Good Reduction of Abelian Varieties

Related

• An obstruction to lifting to characteristic 0
• Global Frobenius Liftability I
• Some elementary examples of non-liftable schemes
• Quasi-canonical lifting of projective varieties in positive characteristic
• Lifts of line bundles on curves on K3 surfaces