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

Abstract

In this paper we investigate the cone Pseff n ⁡ ( C d ) \operatorname {Pseff}_n(C_d) of pseudoeffective n n -cycles in the symmetric product C d C_d of a smooth curve C C . We study the convex-geometric properties of the cone D n ( C d ) \mathcal {D}_n(C_d) generated by the n n -dimensional diagonal cycles. In particular we determine its extremal rays and we prove that D n ( C d ) \mathcal {D}_n(C_d) is a perfect face of Pseff n ⁡ ( C d ) \operatorname {Pseff}_n(C_d) along which Pseff n ⁡ ( C d ) \operatorname {Pseff}_n(C_d) is locally finitely generated.