N\oe ther decomposition for birational maps

TL;DR

This paper investigates the minimal decomposition of birational maps of the complex projective plane into automorphisms and quadratic maps, establishing bounds on the number of quadratic components based on the map's degree.

Contribution

It provides explicit bounds on the minimal number of quadratic maps needed to decompose any birational map of degree d.

Findings

Lower bound: ig[rac{ ext{ln } d}{ ext{ln } 2}ig]

Upper bound: 2(2d-1)

Decomposition bounds depend logarithmically and linearly on degree d

Abstract

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