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On Algebraic and Non-Algebraic Neighborhoods of Rational Curves
We prove that for any d>0 there exists an embedding of the Riemann sphere P^1 in a smooth complex surface, with self-intersection d, such that the germ of this embedding cannot be extended to an embedding in an algebraic surface but the field of germs of meromorphic functions along C has transcendence degree 2 over the field of complex numbers. We give two different constructions of such neighborhoods, either as blowdowns of a neighborhood of the smooth plane conic, or as ramified coverings of a neighborhood of a hyperplane section of a surface of minimal degree.
The proofs of non-algebraicity of these neighborhoods are based on a classification, up to isomorphism, of algebraic germs of embeddings of P^1, which is also obtained in the paper.