Characterizing Varieties Using Birational Transformations

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.

15 pages