Birational geometry of del Pezzo surfaces of degree 4

TL;DR

The paper proves that for a smooth del Pezzo surface S of degree 4 with Picard rank 1, any del Pezzo surface of degree 4 birational to S is isomorphic to S itself, and it extends this rigidity to equivariant settings under finite group actions. The approach combines the Skorobogatov biregular classification, via data such as the discriminant $ riangle(S)$ and the $G(S)$-torsor of lines, with a detailed analysis of birational models through Sarkisov theory and conic-bundle structures. It develops a robust framework using markings and Weyl-group actions to control birational maps, and then lifts the results to $\Gamma$-equivariant birational geometry, establishing rigidity and super-rigidity in the presence of group actions. The results provide a precise description of all birational models of degree-4 del Pezzo surfaces and lay groundwork for broader equivariant birational rigidity questions in surface theory.

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.

Theorems & Definitions (107)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: BFSZ
• Theorem 1.4
• Theorem 1.5: see Skorobogatov-Kummer
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10