On the structure of the birational Abel morphism (Q580462)

Let C be a smooth curve of genus g over a field k (algebraically closed and of arbitrary characteristic). One has the natural Abel morphism of degree \(g,\) \(\phi_ g: S^ gC\to J_ g(C),\) from the g-th symmetric product to the Jacobian. We study the structure of the birational morphism \(\phi_ g\). Precisely, we prove the following result: The birational morphism \(\phi_ g: S^ gC\to J_ g(C)\) is the blowing- up of \(J_ g(C)\) with respect to the prime ideal \({\mathfrak p}\) defining the subvariety \(W^ *_{g-2}\) of special divisors of degree g. - This result allows us to classify all the birational morphisms \(\phi: S^ gC\to J_{g-1}\) such that \(S^ gC\) has a structure of Hilbert scheme of degree \(g\) over C, \(J_{g-1}\) has a structure of Picard scheme of degree \(g-1\) (with \(W_{g-1}\) as natural polarization) and \(\phi\) is the corresponding Abel morphism. This question is closely related with the approaches to the Torelli theorem and Schottky problem performed by the author.