Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.10447 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929502419943424 |
|---|---|
| author | Lvovski, Serge |
| author_facet | Lvovski, Serge |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10447 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On algebraic and non-algebraic neighborhoods of rational curves Lvovski, Serge Algebraic Geometry 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. |
| title | On algebraic and non-algebraic neighborhoods of rational curves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2301.10447 |