Automorphisms of pointless surfaces

Abstract

For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of birational automorphisms of X has bounded finite subgroups. As a key step in the proof, we show boundedness of finite subgroups in any anisotropic reductive algebraic group over a perfect field that contains all roots of 1. Also, we provide applications to Jordan property for groups of birational automorphisms.

Minor corrections; 46 pages

References in corpus (4)

• Localization of modules for a semisimple Lie algebra in prime characteristic
• Construction of Hilbert and Quot Schemes
• The rationality problem for conic bundles
• Bounds for the orders of the finite subgroups of G(k)

Cited by in corpus (5)

• Boundedness results for singular Fano varieties and applications to Cremona groups
• Finite $p$-groups of birational automorphisms and characterizations of rational varieties
• Automorphism groups of $P^1$-bundles over a non-uniruled base
• Non-abelian groups acting on Severi-Brauer surfaces
• The most symmetric smooth cubic surface