Group theoretical characterizations of rationality

Abstract Let $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> be an irreducible variety and $\operatorname{Bir}(X)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>Bir</mml:mo> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:math> its group of birational transformations. We show that the group structure of $\operatorname{Bir}(X)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>Bir</mml:mo> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:math> determines whether $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is rational and whether $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is ruled. Additionally, we prove that any Borel subgroup of $\operatorname{Bir}(X)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>Bir</mml:mo> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:math> has derived length at most twice the dimension of $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , with equality occurring if and only if $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is rational and the Borel subgroup is standard. We also provide examples of non-standard Borel subgroups of $\operatorname{Bir}(\mathbb{P}^{n})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>Bir</mml:mo> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:math> and $\operatorname{Aut}(\mathbb{A}^{n})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>Aut</mml:mo> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:math> , thereby resolving conjectures by Popov and Furter-Poloni.