Effective cycles on the symmetric product of a curve, II: The Abel–Jacobi faces

In this paper, we study the convex-geometric properties of the cone of pseudoeffective n-cycles in the symmetric product C_d of a smooth curve C. We introduce and study the Abel-Jacobi faces, related to the contractibility properties of the Abel-Jacobi morphism and to classical Brill-Noether varieties. We investigate when Abel-Jacobi faces are non-trivial, and we prove that for d sufficiently large (with respect to the genus of C) they form a maximal chain of perfect faces of the tautological pseudoeffective cone (which coincides with the pseudoeffective cone if C is a very general curve).