General birationality and Hyperelliptic Theta divisors

Author:

Fabrizio Catanese

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Complex Variables (math.CV)

journal:

date:

2023-10-13 16:00:00

Abstract

We first show that the condition of having a birational map onto the image is an open condition for irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of rational maps, via specializations. As an application, we give 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 $\Phi_X$ 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.

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