Complex algebraic curves. Annuli

Details Complex algebraic curves. Annuli Complex algebraic curves. Annuli

2007-08-13

arxiv.org/abs/0708.1661v1

Mathematics

Algebraic Geometry

43 pages. This is the full, unabridged version of our article "Complex algebraic curves via Poincare--Hopf formula. II. Annuli

Scientific paper

We provide the full classification of algebraic embeddings of $\mathbb{C}^*$ into $\mathbb{C}^2$ satisfying certain regularity condition, which conjecturally holds for all algebraic maps from $\mathbb{C}^*$ into $\mathbb{C}^2$. The resulting list comprises 1 smooth family, 18 discrete families and 4 special cases. Any embedding known to us can be reduced to one of this list by a de Jonqui\`ere transform and a suitable change of variables. The classification uses in general tools from previous work "Complex algebraic curves via Poincare--Hopf formula. I. Parametric lines." (Pacific. J. Math. 229 (2007) No. 2, 307--338): we carefully estimate Milnor numbers of singularities that may appear in the embedding of $\mathbb{C}^*$. We use the regularity condition to bound the sum of so--called codimensions of singular points. The detailed discussion of this condition can be found in http://www.mimuw.edu.pl/~mcboro/pliki/artykuly/curv4.pdf

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