0708.1661Aug 13, 200743 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
We provide the full classification of algebraic embeddings of C∗ into C2 satisfying certain regularity condition, which conjecturally holds for all algebraic maps from C∗ into C2
Nearby in the stack
. 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
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