Birational geometry of del Pezzo surfaces of degree 4

2025/12/22 by Constantin Aleksandrovich Shramov, Shramov, Constantin, Andrey Trepalin +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · doi:10.48550/arxiv.2512.19660

openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

Abstract

It is known that any Mori fiber space birational to a minimal smooth del Pezzo surface S of degree 4 is either a del Pezzo surface of degree 4 itself, or a smooth cubic surface with a structure of a relatively minimal conic bundle. We show that any del Pezzo surface of degree 4 birational to S is actually isomorphic to S. Also, we sketch an equivariant version of this fact. On the way, we review the biregular classification of del Pezzo surfaces of degree 4 obtained by A. N. Skorobogatov.

Citations

• Atomic decompositions for derived categories of G-surfaces
• Birational Geometry of sextic del Pezzo surfaces
• Automorphisms of del Pezzo surfaces without points
• Geometry and arithmetic of geometrically integral regular del Pezzo surfaces
• Explicit Sarkisov program for regular surfaces over arbitrary fields and applications
• Birational classification of pointless del Pezzo surfaces of degree 8
• Minimal model program for excellent surfaces
• Lie groups and Lie algebras

Related

• Birational classification of pointless del Pezzo surfaces of degree 8
• Birational Geometry of sextic del Pezzo surfaces
• Birational rigidity of orbifold degree 2 del Pezzo fibrations
• Classification of del Pezzo surfaces of rank one. I. Height 1 and 2. II. Descendants with elliptic boundaries
• G-birational superrigidity of del Pezzo surfaces of degree 2 and 3