5 February 2025 — Padova

A characterization of rationality and Borel subgroups in the Cremona group

Andriy Regeta, University of Jena

Abstract

In this talk I will present the following two results about the group of birational transformations (which we denote by \(\mathsf{Bir}(X)\)) of an irreducible variety \(X\): The first one: If \(\mathbb{P}^n\) is a projective space and if \(\mathsf{Bir}(X)\) is isomorphic to \(\mathsf{Bir}(\mathbb{P}^n)\), than \(X\) is rational. Regarding the second one: it is well-known that Borel subgroups of an algebraic group (over algebraically closed fields) are conjugate. This is not the case for \(\mathsf{Bir}(\mathbb{P}^2)\). Nevertheless, all Borel subgroups of \(\mathsf{Bir}(\mathbb{P}^2)\) were classified by J.-P. Furter and I. Heden. In the general case we show that a Borel subgroup of \(\mathsf{Bir}(X)\) has derived length at most twice the dimension of \(X\) and in the case of equality \(X\) is rational, and the Borel subgroup is conjugate to the standard Borel subgroup in \(\mathsf{Bir}(\mathbb{P}^n)\). Moreover, we provide examples of Borel subgroups in \(\mathsf{Bir}(\mathbb{P}^n)\) of derived length strictly smaller than \(2n\) for any \(n\leq2\). This confirms the conjectures by Popov and by Furter-Heden. We also show that the automorphism group \(\mathsf{Aut}(\mathbb{A}^n)\) of an affine space \(\mathbb{A}^n\) contains non-conjugate Borel subgroups of \(n>2\) and provide some structure results concerning the Borel subgroups of \(\mathsf{Aut}(\mathbb{A}^n)\). This talk is based on joint work with Christian Urech and Immanuel van Santen.