Effective cycles on the symmetric product of a curve, I: the diagonal cone. (with an appendix by Ben Moonen)

In this paper and in its sequel [BKLV], we investigate the cone ${\rm Pseff}_n(C_d)$ of pseudoeffective $n$-cycles in the symmetric product $C_d$ of a smooth curve $C$. In the present paper, we study the convex-geometric properties of the cone generated by the $n$-dimensional diagonal cycles, which we call the $n$-dimensional diagonal cone. We prove that the $n$-dimensional diagonal cone is a perfect face of ${\rm Pseff}_n(C_d)$ along which ${\rm Pseff}_n(C_d)$ is locally finitely generated.Comment: v1: 42 pages, 2 figures. v2: added remark 2.10, an appendix by Ben Moonen and changed accordingly the title; separated the bibliography of the paper from the one of the appendices; 46 pages, 2 figures. v3: removed some parts of appendix A, simplified proof of Prop. A.9(d), shortened the introduction; 41 pages, 2 figures. To appear on Transactions AM