In this paper, which is a sequel of [BKLV17], we study the convex-geometric properties of the cone of pseudoeffective -cycles in the symmetric product of a smooth curve . We introduce and study the Abel–Jacobi faces, related to the contractibility properties of the Abel–Jacobi morphism and to classical Brill–Noether varieties. We investigate when Abel–Jacobi faces are non-trivial, and we prove that for sufficiently large (with respect to the genus of ) they form a maximal chain of perfect faces of the tautological pseudoeffective cone (which coincides with the pseudoeffective cone if is a very general curve).
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Francesco Bastianelli, Alexis Kouvidakis, Angelo Felice Lopez, Filippo Viviani, Effective cycles on the symmetric product of a curve, II: the Abel–Jacobi faces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 4, pp. 839–878DOI 10.4171/RLM/917