Effective cycles on the symmetric product of a curve, II: the Abel-Jacobi faces

TL;DR

This paper explores the geometric structure of the cone of pseudoeffective cycles in symmetric products of curves, introducing Abel-Jacobi faces and analyzing their properties, especially for large degree and very general curves.

Contribution

It introduces Abel-Jacobi faces related to the Abel-Jacobi morphism and characterizes their structure in the pseudoeffective cone of symmetric products of curves.

Findings

Abel-Jacobi faces form a maximal chain of perfect faces for large degree.

For very general curves, the pseudoeffective cone coincides with the tautological cone.

Non-trivial Abel-Jacobi faces are characterized by the degree and genus of the curve.

Abstract

In this paper, which is a sequel of [BKLV], we study the convex-geometric properties of the cone of pseudoeffective $n$-cycles in the symmetric product $C_d$ of a smooth curve $C$. 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 $d$ sufficiently large (with respect to the genus of $C$) they form a maximal chain of perfect faces of the tautological pseudoeffective cone (which coincides with the pseudoeffective cone if $C$ is a very general curve).