The geometry of sporadic C⁎C⁎-embeddings into C2C2

A closed algebraic embedding of C⁎=C1∖{0}C⁎=C1∖{0} into C2C2 is sporadic   if for every curve A⊆C2A⊆C2 isomorphic to an affine line the intersection with C⁎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)→(P2,U)(X,D)→(P2,U), where U   is the closure of C⁎C⁎ on P2P2. We show in particular that one can choose coordinates on C2C2 in which the type at infinity of the C⁎C⁎ and the self-intersection of its proper transform on X are sharply limited.