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

TL;DR

This paper explores the geometric structure of the cone of pseudoeffective cycles in symmetric products of curves, focusing on the diagonal cycles and revealing their role as a perfect face of this cone.

Contribution

It introduces and analyzes the n-dimensional diagonal cone, establishing its status as a perfect face of the pseudoeffective cone in symmetric products of curves.

Findings

The n-dimensional diagonal cone is a perfect face of the pseudoeffective cone.

The pseudoeffective cone is locally finitely generated along the diagonal cone.

The study provides geometric insights into the structure of effective cycles on symmetric products.

Abstract

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.