Complex algebraic curves. Annuli
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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 Jonquière 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
43 pages. This is the full, unabridged version of our article "Complex algebraic curves via Poincare--Hopf formula. II. Annuli". In this version we include all detailed estimates. The TeX file has been prepared using Scientific Workplace
43 pages. This is the full, unabridged version of our article "Complex algebraic curves via Poincare--Hopf formula. II. Annuli". In this version we include all detailed estimates. The TeX file has been prepared using Scientific Workplace