On symmetric products of curves

TL;DR

This paper investigates the extension of gonality concepts to the symmetric product of a curve, establishing bounds on irrationality degree and gonality, and exploring geometric properties related to correspondences and linear subspaces.

Contribution

It provides new bounds on the degree of irrationality and gonality for symmetric products of generic curves, using correspondences and monodromy techniques.

Findings

Degree of irrationality of X is at least g-1 for generic C.

Minimum gonality of curves through a generic point of X equals gonality of C.

New bounds on the ample cone for certain genus ranges.

Abstract

Let C be a smooth complex projective curve of genus g and let X be its second symmetric product. This paper concerns the study of some attempts at extending to X the notion of gonality. In particular, we prove that the degree of irrationality of X is at least g-1 when C is a generic curve, and that the minimum gonality of curves through the generic point of X equals the gonality of C. In order to produce the main results we deal with correspondences on the k-fold symmetric product of C, with some interesting linear subspaces of \mathbb{P}^n enjoying a condition of Cayley-Bacharach type, and with monodromy of rational maps. As an application, we also give new bounds on the ample cone of X when C is a generic curve of genus 5<g<9.