2007/08/13 by Maciej Borodzik, Borodzik, Maciej, Henryk Zoladek +1
Mathematics · #14H45 #14H50 #14R05 (Primary) #32S05 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H45 #msc:14H50 #msc:14R05 #msc:32S05
paper · pdf · doi:10.48550/arxiv.0708.1661
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
arxiv created 2007/08/13 · arxiv updated 2009/12/01
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. 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