Birational geometry of del Pezzo surfaces of degree 4

TL;DR

This paper proves that del Pezzo surfaces of degree 4 are birationally rigid, meaning any birational map to such a surface is an isomorphism, and discusses related classifications and equivariant versions.

Contribution

It establishes the birational rigidity of degree 4 del Pezzo surfaces and reviews their biregular classification, including an equivariant perspective.

Findings

Any Mori fiber space birational to a degree 4 del Pezzo surface is either the same surface or a cubic surface with a conic bundle structure.

Any del Pezzo surface of degree 4 birational to a given surface is actually isomorphic to it.

Provides an overview of the biregular classification of degree 4 del Pezzo surfaces.

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.