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A closed algebraic embedding of C ⁎ = C 1 ∖ { 0 } into C 2 is sporadic if for every curve A ⊆ C 2 isomorphic to an affine line the intersection with C ⁎ is at least 2. Non-sporadic embeddings have been classified. There are very few known sporadic embeddings. We establish geometric and algebraic tools to classify them based on the analysis of the minimal log resolution ( X , D ) → ( P 2 , U ) , where U is the closure of C ⁎ on P 2 . We show in particular that one can choose coordinates on C 2 in which the type at infinity of the C ⁎ and the self-intersection of its proper transform on X are sharply limited.

We consider closed curves C≃C∗ in the affine plane S≃C2 that admit a good or very good asymptote. By this we mean a curve L≃C in S that in suitable coordinates for S is linear and tangent to C at infinity, and meets C once or not at all at finite distance. We classify these curves up to automorphism of S. Relying on the theory of open algebraic surfaces we first determine the possibilities for the singularities of C at infinity and then proceed to give explicit equations.

The purpose of this paper is to present an algorithm for computing all the asymptotes of a real plane algebraic curve. By this algorithm, all the asymptotes of a real plane algebraic curve may be represented via polynomial real root isolation.