2024/12/30 by Karol Palka, Palka, Karol, Tomasz Pełka +1
Engineering · Mathematics · #14J10 (Primary) 14D06 #14J45 #14R05 (Secondary) #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2412.21174
openalex publication_date 2024/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is the first article in a series aimed at classifying normal del Pezzo surfaces of Picard rank one over algebraically closed fields of arbitrary characteristic up to an isomorphism. Our guiding invariant is the height of a del Pezzo surface, defined as the minimal intersection number of the exceptional divisor of the minimal resolution and a fiber of some ℙ1-fibration. The geometry of del Pezzo surfaces gets more constrained as the height grows; in characteristic 0 no example of height bigger than 4 is known. In this article, we classify del Pezzo surfaces of Picard rank one and height at most 2; in particular we describe the non-log terminal ones. We also describe a natural class of del Pezzo surfaces which have descendants with elliptic boundary, i.e. whose minimal resolution has a birational morphism onto a canonical del Pezzo surface of rank one mapping the exceptional divisor to an anti-canonical curve.