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 p...

We give characterizations of contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. From this we obtain some generalizations of the classical criteria for contractibility of Castelnuovo and Artin. Furthermore, we will derive a finiteness result on homogeneous spectra defined by Weil divisors on proper normal algebraic surfaces.