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In this paper and in its sequel [BKLV], we investigate the cone ${\rmPseff}_n(C_d)$ of pseudoeffective $n$-cycles in the symmetric product $C_d$ ofa smooth curve $C$. In the present paper, we study the convex-geometricproperties of the cone generated by the $n$-dimensional diagonal cycles, whichwe call the $n$-dimensional diagonal cone. We prove that the $n$-dimensionaldiagonal cone is a perfect face of ${\rm Pseff}_n(C_d)$ along which ${\rmPseff}_n(C_d)$ is locally finitely generated.

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