Curves of genus \(g\) lying on a \(g\)-dimensional Jacobian variety
type: Item
instance of: scholarly article
title: Curves of genus \(g\) lying on a \(g\)-dimensional Jacobian variety
zbMATH Open document ID: 0638.14025
DOI: 10.1007/BF01393898
author: Fabio Bardelli
author: Gian Pietro Pirola
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143653
review text: The authors prove the following theorem: Let \(J\) be a general Jacobian variety of dimension \(g\geq 4\) defined over \({\mathbb{C}}\). If \(\chi: J'\to J\) is an isogeny and J' is a Jacobian then there is an isomorphism of principally polarized abelian varieties \(\mu: J\to J'\) and \(\chi\circ \mu\) is the multiplication by an integer. The proof is based on a \((3g-3)\)-variables degeneration of \(J\) to a generalized Jacobian of an irreducible stable curve with two nodes, on the analysis of the extension classes coming from each node and on the comparison of the Gauss maps associated to the two surfaces of extension classes coming from the corresponding degenerations of \(J\) and of \(J'\) respectively. If \(J\) is the Jacobian \(J(C)\) of a smooth curve \(C\), \(\Phi: C\to J\) is some fixed Abel Jacobi embedding, and \(\Phi _ n: C\to J\) is the map \(n\circ \Phi\) the theorem above has the following corollary: The irreducible curves of geometric genus \(g\) contained in a general Jacobian \(J(C)\) of dimension \(g\geq 4\) are the curves \(\Phi _ n(C)\) and all their translates. In particular they are all birationally equivalent.
Mathematics Subject Classification ID: 14K30
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 4040084
zbMATH Keywords: general Jacobian variety of dimension \(g\geq 4\)
zbMATH Keywords: generalized Jacobian of an irreducible stable curve with two nodes
zbMATH Keywords: isogeny
zbMATH Keywords: geometric genus
MaRDI profile type: Publication
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OpenAlex ID: W2327927342
description: scientific article; zbMATH DE number 4040084