Abstract. We give the outline of the proof of the linearization conjecture: every algebraic C * -action on C 3 is linear in a suitable coordinate system.
Let E ⊆ P 2 be a complex rational cuspidal curve contained in the projective plane. The Coolidge-Nagata conjecture asserts that E is Cremona equivalent to a line, i.e. it is mapped onto a line by some birational transformation of P 2 . In [Pal14a] the second author analyzed the log minimal model program run for the pair (X, 1 2 D), where (X, D) → (P 2 , E) is a minimal resolution of singularities, and as a corollary he established the conjecture in case when more than one irreducible curve in P 2 \ E is contracted by the process of minimalization. We prove the conjecture in the remaining cases.
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.
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