On the cone of effective cycles on the symmetric products of a curve

I will report on a joint work with F. Bastianelli, A. Kouvidakis and A. F. Lopez in which we study the cone of (pseudo-)effective cycles on symmetric products of a curve. We first prove that the diagonal
cycles span a face of the pseudo-effective cone of cycles in any given dimension. Secondly, we look at the contractibility faces associated to the Abel-Jacobi morphism towards the Jacobian and in many cases
we are able to compute their dimension. The geometry of linear series of a curve (e.g. the classical Brill-Noether theory) will play a special role in our analysis.