General birationality and Hyperelliptic Theta divisors
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917651958202368 |
|---|---|
| author | Catanese, Fabrizio |
| author_facet | Catanese, Fabrizio |
| contents | We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of rational maps, via specializations.
Among the applications is a new proof of a result obtained jointly with Luca Cesarano: that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $Φ_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal.
The proof is also based on a careful study of the Theta divisors of the Jacobians of Hyperelliptic curves, and some related geometrical constructions. We investigate these here also in view of their beauty and of their independent interest, as they lead to a description of the rings of Hyperelliptic theta functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09548 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | General birationality and Hyperelliptic Theta divisors Catanese, Fabrizio Algebraic Geometry Complex Variables 14E05, 14E25, 14M99, 14K25, 14K99, 14H40, 32J25, 32Q55, 32H04 We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of rational maps, via specializations. Among the applications is a new proof of a result obtained jointly with Luca Cesarano: that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $Φ_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal. The proof is also based on a careful study of the Theta divisors of the Jacobians of Hyperelliptic curves, and some related geometrical constructions. We investigate these here also in view of their beauty and of their independent interest, as they lead to a description of the rings of Hyperelliptic theta functions. |
| title | General birationality and Hyperelliptic Theta divisors |
| topic | Algebraic Geometry Complex Variables 14E05, 14E25, 14M99, 14K25, 14K99, 14H40, 32J25, 32Q55, 32H04 |
| url | https://arxiv.org/abs/2310.09548 |