Autour d'une surface rationnelle dans $\mathbb{C}^3$

Details Autour d'une surface rationnelle dans $\mathbb{C}^3$ Autour d'une surface rationnelle dans $\mathbb{C}^3$

2005-12-07

arxiv.org/abs/math/0512152v1

Mathematics

Algebraic Geometry

This article is written in an expositional style and focuses on examples. It could also serve as an introduction to recent res

Scientific paper

Affine surfaces in $\mathbb{C}^{3}$ defined by an equation of the form $x^{n}z-Q(x,y)=0$ have been increasingly studied during the past 15 years. Of particular interest is the fact that they come equipped with an action of the additive group $\mathbb{C}_{+}$ induced by such an action on the ambient space. The litterature of the last decade may lead one to believe that there are essentially no other of rational surfaces in $\mathbb{C}^{3}$ with this property. In this note, we construct an explicit example of a surface nonisomorphic to a one of the above type but equipped with a free $\mathbb{C}_{+}$-action induced by an action on $\mathbb{C}^{3}$. We give an elementary and self-contained proof of this fact. As an application, we construct a wild but stably-tame automorphisme of $\mathbb{C}^{3}$ which seems to be new.

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