Nœther decomposition for birational maps

Abstract

Let $\varphi$ be a birational map of the complex projective plane. We know that $\varphi$ can be written as a composition of automorphisms of ${\mathbb{P}}_{\mathbb{C}}^2$ and the standard quadratic birational map $\sigma$. This writing, that is non-unique, is minimal if the number $\mathfrak{n}(\varphi )$ of $\sigma$ is as small as possible. We prove that if $\varphi$ is of degree $d\ge 2$, then $\left[\frac{\ln d}{\ln 2}\right]\le \mathfrak{n}(\varphi )\le 2(2d-1)$.

This paper has been withdrawn by the author due to an error in the upper bound