Characterizing Varieties Using Birational Transformations

TL;DR

This paper investigates how the group of birational transformations characterizes algebraic varieties, showing it uniquely determines ruled varieties but not non-uniruled ones.

Contribution

It establishes that for ruled varieties, the birational transformation group uniquely identifies the variety, unlike for non-uniruled varieties where this does not hold.

Findings

Bir(X) determines ruled varieties up to birational equivalence.

This property fails for non-uniruled varieties.

The group structure encodes key geometric properties.

Abstract

Suppose $X$ is an irreducible complex variety. We show that when $X$ is ruled, the group of birational transformations $Bir(X)$, as a group, determines $X$ up to birational transformations and automorphisms of the base field. In contrast, we demonstrate that this same property never holds for non-uniruled varieties.