Cremona transformations, surface automorphisms and plane cubics

Abstract

We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the group structure on the cubic to understand when the indeterminacy and exceptional behavior of the transformation may be eliminated by repeated blowing up.

29 pages. Many changes, including a new title and the addition of an appendix (contributed by Igor Dolgachev) which carefully treats the group law for singular and reducible cubics. To appear in the Michigan Math Journal

References in corpus (2)

• On the inertia group of elliptic curves in the Cremona group of the plane
• Sur un théorème de Castelnuovo

Cited by in corpus (7)

• Dynamical degrees of birational transformations of projective surfaces
• Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces
• Finiteness of Klein actions and real structures on compact hyperkähler manifolds
• Real structures on rational surfaces and automorphisms acting trivially on Picard groups
• Infinitesimal deformations of rational surface automorphisms
• Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy
• Siegel Disks on Rational Surfaces