Moscow Mathematical Journal
Volume 24, Issue 3, July–September 2024 pp. 391–405.
On Algebraic and Non-Algebraic Neighborhoods of Rational Curves
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
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:
Keywords: Neighborhoods of rational curves, surfaces of minimal degree, blowup.
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