On algebraic and non-algebraic neighborhoods of rational curves

2023.

Lvovsky S.

We prove that for any d>0 there exists an embedding of the Riemann sphere ℙ1in 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 ℂ. 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 ℙ1, which is also obtained in the paper.

Keywords: rational curves

Publication based on the results of: Motivic, categoric and classical algebraic geometry, and its connection to differential geometry of special manifolds (2023)