  In this paper, which is a sequel of [BKLV], 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).
