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

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