Dynamical degrees of birational transformations of projective surfaces

Abstract

ABSTRACT. The dynamical degree λ( f ) of a birational transformation f measures the exponential growth rate of the degree of the formulae that define the n-th iterate of f . We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of λ( f ) and the structure of the conjugacy class of f . For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.

Non simplicité du groupe de Cremona sur tout corps

“…Cependant, même si nous pouvions enlever cette hypothèse et obtenir ainsi une démonstration alternative de la non simplicité de Bir(P 2 ) sur tout corps, une telle preuve reste peu élémentaire car elle repose lourdement sur les articles [7] et [3].…”

“…Ainsi cet élément particulier joue le même rôle que l'élément général a ci-dessus. La preuve présentée dans cet article ne repose pas sur [3,7] à l'exception du lemme 2.8 à la section 2.3 qui est une adaptation directe de [7,Proposition 5.7].…”

Dynamics on supersingular K3 surfaces

“…We point out that the automorphisms in Theorem 3.5 are mostly implicit, i.e. there are existence results building on elliptic fibrations (in particular of maximal rank) and some group theory (see [2,4]). In contrast, until the completion of the first version of this paper it was only for p D 3 that there was an explicit g 2 Aut.X.p// of Salem degree 22 known [3], building on the calculation of Aut.X.3// by Kondō and Shimada [8].…”

“…One of the keys for the implicit results in [2,4] is the special feature that a K3 surface may admit different elliptic fibrations. For convenience, we shall only work with 714 M. Schütt CMH fibrations which are already visible on X .…”

“…Esnault and Oguiso illustrated this phenomenon explicitly with the Fermat quartic in characteristic 3, building on work of Kondō and Shimada [8]. On the implicit side, there are nonliftability results for the supersingular K3 surface X.p/ of Artin invariant D 1 in characteristic p ¤ 5; 7; 13 in [2,4]. Our aim is to exhibit non-liftable automorphisms on an infinite series of supersingular K3 surfaces in an explicit and systematic manner: The key ingredient of our approach consists in the theory of elliptic fibrations on K3 surfaces and in particular Mordell-Weil lattices, since these are at the same time accessible, versatile and allow for explicit descriptions of automorphisms.…”