Effective cycles on the symmetric product of a curve, I: the diagonal cone

In this paper we investigate the cone $Pseff_n(C_d)$ of pseudoeffective $n$-cycles in the symmetric product $C_d$ of a smooth curve $C$. We study the convex-geometric properties of the cone $D_n(C_d)$ generated by the $n$-dimensional diagonal cycles. In particular we determine its extremal rays and we prove that $D_n(C_d)$ is a perfect face of $Pseff_n(C_d)$ along which $Pseff_n(C_d)$ is locally finitely generated.