Characterizing Varieties Using Birational Transformations
Nathan Chen, Louis Esser, Andriy Regeta, Christian Urech, Immanuel van Santen
TL;DR
The paper investigates when the birational transformation group $\mathrm{Bir}(X)$ determines an irreducible complex variety $X$. It proves a strong positive result for ruled varieties: if $\mathrm{Bir}(X) \cong \mathrm{Bir}(\mathbb{A}^s \times Y)$, then $X$ is birational to $\mathbb{A}^s \times Z$ with $\mathbb{C}(Z)$ isomorphic to $\mathbb{C}(Y)$ up to a ${\mathbb C}$-automorphism, and some birational models are isomorphic over ${\mathbb Z}$. A key technique is showing that group isomorphisms preserve unipotent elements, which enables reconstruction of translation subgroups and, ultimately, a field isomorphism. The paper also demonstrates a clear limitation: for irreducible non-uniruled varieties $X$, the map $\mathrm{Bir}(X) \to \mathrm{Bir}(X\times C)$ is an isomorphism for very general curves $C$ of high genus, illustrating that $\mathrm{Bir}(X)$ cannot always determine $X$. Together, these results delineate when birational groups are complete invariants and raise questions for the uniruled-but-not-ruled case and base-field automorphism behavior.
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.