Curves of genus \(g\) lying on a \(g\)-dimensional Jacobian variety
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.
general Jacobian variety of dimension \(g\geq 4\)
generalized Jacobian of an irreducible stable curve with two nodes
scientific article; zbMATH DE number 4040084