A characterization of double covers of curves in terms of the ample cone of second symmetric product

TL;DR

This paper studies the nef cone of the second symmetric product of a curve, providing a characterization of double covers of curves of certain genus, extending previous results by Debarre.

Contribution

It generalizes Debarre's result by characterizing double covers of curves via the nef cone of their second symmetric product for a broader genus range.

Findings

The nef cone is spanned by diagonal and fiber classes.

Double covers are characterized for genus up to (g-1)/8.

Extends previous work by Debarre.

Abstract

We investigate the nef cone spanned by the diagonal and the fibre classes of second symmetric product of a curve of genus $g$. This 2-dimensional nef cone gives a characterization of double covers of curves of genus $\le \frac{g-1}{8}$. This is a generalization of a result by Debarre.