Moscow Mathematical Journal

Volume 24, Issue 3, July–September 2024 pp. 391–405.

On Algebraic and Non-Algebraic Neighborhoods of Rational Curves

Authors: Serge Lvovski (1)
Author institution:(1) National Research University Higher School of Economics, Russian Federation;
Federal State Institution “Scientific-Research Institute for System Analysis of the Russian Academy of Sciences” (SRISA)

Summary:

We prove that for any $d>0$ there exists an embedding of the Riemann sphere $\mathbb{P}^1$ 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 $\mathbb{C}$ . 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 $\mathbb{P}^1$ , which is also obtained in the paper.

2020 Math. Subj. Class. 32H99, 14J26

Keywords: Neighborhoods of rational curves, surfaces of minimal degree, blowup.

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