Stéphane Lamy

Normal subgroups in the Cremona group

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane

Normal subgroup generated by a plane polynomial automorphism

\mathbb{P}_{\mathbf{k}}^2

On the Genus of Birational Maps Between Threefolds

is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory and algebraic geometry to produce elements in the Cremona group that generate non-trivial normal subgroups.

The tame and the wild automorphisms of an affine quadric threefold

We characterise smooth curves in P3 whose blow-up produces a threefold with anticanonical divisor big and nef. These are curves C of degree d and genus g lying on a smooth quartic, such that (i) 4d

Birational Self-maps and Piecewise Algebraic Geometry

We study the normal subgroup 〈f〉N generated by an element f ≠ id in the group G of complex plane polynomial automorphisms having Jacobian determinant 1. On the one hand, if f has length at most 8 relative to the classical amalgamated product structure of G, we prove that 〈f〉N = G. On the other hand, if f is a sufficiently generic element of even length at least 14, we prove that 〈f〉N ≠ G.

THE TAME AUTOMORPHISM GROUP OF AN AFFINE QUADRIC THREEFOLD ACTING ON A SQUARE COMPLEX

We describe the structure of the group of algebraic automorphisms of the following surfaces 1) P1 k ×P1 k minus a diagonal; 2) P1 k ×P 1 k minus a fiber. The motivation is to get a new proof of two theorems proven respectively by L. Makar-Limanov and H. Nagao. We also discuss the structure of the semi-group of polynomial proper maps from C2 to C2.

Groupes d’automorphismes polynomiaux du plan

In this note we present two equivalent definitions for the genus of a birational map \(\varphi: X --\rightarrow Y\) between smooth complex projective threefolds. The first one is the definition introduced by Frumkin [Mat. Sb. (N.S.) 90(132):196–213, 325, 1973], and the second one was recently suggested to me by S. Cantat. By focusing first on proving that these two definitions are equivalent, one can obtain all the results in M.A. Frumkin [Mat. Sb. (N.S.) 90(132):196–213, 325, 1973] in a much shorter way. In particular, the genus of an automorphism of \(\mathbb{C}^{3}\), view as a birational self-map of \(\mathbb{P}^{3}\), will easily be proved to be 0.

Automorphismes polynomiaux préservant une action de groupe

We prove the existence of wild automorphisms on an affine quadric threefold. The method we use is an adaptation of the one used by Shestakov and Umirbaev to prove the existence of wild automorphisms on the affine three dimensional space.