[Paper Review] Complex algebraic curves. Annuli

Maciej Borodzik, Henryk Żołądek|ArXiv.org|Aug 13, 2007

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper classifies complex algebraic curves in ℂ² that are homeomorphic to ℂ* (the punctured complex plane), under a regularity condition on Puiseux expansions at singularities and infinity. Using Poincaré–Hopf index theory and invariant estimation for vector fields, the authors provide a complete list of 19 families and 4 exceptional cases, up to polynomial diffeomorphisms and reparametrizations, with explicit parametric forms and singularity types identified.

ABSTRACT

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

Motivation & Objective

• To classify affine algebraic curves in ℂ² that are topologically equivalent to ℂ* (i.e., embedded annuli) under natural regularity conditions.
• To identify all such curves up to equivalence under polynomial diffeomorphisms and reparametrizations of the parameter.
• To characterize the singularities (finite and at infinity) and asymptotic behavior of these curves via Puiseux expansions and invariant counting.
• To establish a complete list of non-equivalent curves satisfying the regularity condition, using index-theoretic and algebraic methods.

Proposed method

• Apply the Poincaré–Hopf formula to a carefully chosen vector field on the curve to compute the sum of hidden double point invariants at singularities and infinity.
• Estimate the number of double points hidden at singularities and at infinity using Puiseux expansions and their regularity conditions.
• Use the Jung–van der Kulk theorem to reduce equivalence to compositions of linear maps and elementary transformations (x+P(y), y), (x, y+Q(x)).
• Employ recursive Laurent polynomial constructions (e.g., R_{k,m}, S_k, T_k, etc.) to generate families of curves with controlled asymptotic and singularity behavior.
• Impose a regularity condition on Puiseux coefficients to ensure the invariants are well-behaved and the classification is complete.
• Use case analysis based on degree and valuation parameters (p, q, r, s, m, n) to eliminate non-realizable configurations and prove strictness of the classification.

Experimental results

Research questions

• RQ1 Which algebraic curves in ℂ² are topologically equivalent to ℂ* and satisfy the regularity condition on Puiseux coefficients at singularities and infinity?
• RQ2 How can such curves be completely classified up to polynomial diffeomorphism and reparametrization?
• RQ3 What are the precise parametric forms of all non-equivalent algebraic annuli satisfying the regularity condition?
• RQ4 How do the invariants of double points hidden at singularities and infinity combine to satisfy the Poincaré–Hopf formula?
• RQ5 Which configurations of singularities (e.g., An-type) and asymptotic behaviors are realizable under the regularity condition?

Key findings

• All algebraic annuli satisfying the regularity condition are equivalent to one of 19 parametric families or 4 exceptional cases listed in the Main Theorem.
• The classification is complete and pairwise non-equivalent under polynomial diffeomorphisms and reparametrizations of the parameter t.
• The curves in family (a) are smooth and have two points at infinity if n ≤ m, otherwise one; they are rational curves with monomial leading terms.
• Families (b)–(f) exhibit A_{2m} singularities at finite points (e.g., at t=1/2 or t=1), while others are smooth but have non-trivial asymptotic behavior at infinity.
• The exceptional cases (k)–(w) include curves with higher-order singularities such as A_6 or multiple A_2 points, and are distinguished by specific parameter constraints and symmetry.
• The proof shows that only specific parameter values (e.g., α=2, β=1/2 in case (w)) yield embedded annuli, excluding other configurations due to incompatible singularity structures.