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Let X be a non-singular rational algebraic surface (over C) equipped with a
birational morphism to CP^2. Let L be a line on CP^2, and E' be the union of
the strict transform of L with all but one irreducible component of the
exceptional divisor. Assume that the matrix of intersection numbers of
components of E' is negative definite, so that E' can be contracted to a
normal analytic surface X'. Question: When is X' algebraic? I will present
an answer in a geometric and then an algebraic form, and sketch the idea of
the proof.
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