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Seminar of the Laboratory on Algebraic Transformation Groups HSE University
September 11, 2024 18:00, Moscow, online, for the zoom link contact dchunaev@hse.ru


A Characterization of Rationality and Borel Subgroups

Immanuel van Santen

Universität Basel


https://youtu.be/k8gduDNqmrI

Abstract: This is joint work with Andriy Regeta and Christian Urech. In this talk, we focus on the following two questions about the group of birational transformations, $\mathrm{Bir}(X)$, of an irreducible variety $X$:
1. If $\mathrm{Bir}(X)$ and $\mathrm{Bir}(\mathbb{P}^n)$ are isomorphic, does this imply that $X$ and $\mathbb{P}^n$ are birational?
2. What are the Borel subgroups of $\mathrm{Bir}(X)$?

The first question was answered affirmatively in 2014 by Serge Cantat under the additional assumption that $\mathrm{dim} X \leq n$. We prove that the first question has an affirmative answer without this extra assumption (and we do not use the result of Serge Cantat).
Regarding the second question, Jean-Philippe Furter and Isac Hedén completely classified the Borel subgroups of $\mathrm{Bir}(\mathbb{P}^n)$ in 2023 for the case $n = 2$. We prove that any Borel subgroup of $\mathrm{Bir}(X)$ has derived length at most twice the dimension of $X$, and if equality holds, then $X$ is rational, and the Borel subgroup is conjugate to the standard Borel subgroup in $\mathrm{Bir}(\mathbb{P}^n)$. Moreover, we provide examples of Borel subgroups in $\mathrm{Bir}(\mathbb{P}^n)$ of derived length less than $2n$ for any $n \geq 2$ (the case $n = 2$ was treated by Furter and Hedén). This answers affirmatively a conjecture of Vladimir Popov.


© Steklov Math. Inst. of RAS, 2024