Moving curves of least gonality on symmetric products of curves

TL;DR

This paper investigates the minimal gonality of curves within symmetric products of a smooth projective curve, establishing that under certain conditions, the only such curves are translates of the original curve, and comparing covering and connecting gonality.

Contribution

It proves that for general curves, the only curves computing covering gonality in symmetric products are translates of the original curve, and shows connecting gonality exceeds covering gonality.

Findings

Covering gonality equals the gonality of C for certain symmetric products.

Only curves of the form C+p compute the covering gonality under mild assumptions.

Connecting gonality is strictly larger than covering gonality.

Abstract

This paper is a sequel of arXiv:2208.00990. Let $C$ be a smooth complex projective curve of genus $g$ and let $C^{(k)}$ be its $k$-fold symmetric product. The covering gonality of $C^{(k)}$ is the least gonality of an irreducible curve $E\subset C^{(k)}$ passing through a general point of $C^{(k)}$. It follows from previous works of the authors that if $2\leq k\leq 4$ and $g\geq k+4$, the covering gonality of $C^{(k)}$ equals the gonality of $C$. In this paper, we prove that under mild assumptions of generality on $C$, the only curves $E\subset C^{(k)}$ computing the covering gonality of $C^{(k)}$ are copies of $C$ of the form $C+p$, for some point $p\in C^{(k-1)}$. As a byproduct, we deduce that the connecting gonality of $C^{(k)}$ (i.e. the least gonality of an irreducible curve $E\subset C^{(k)}$ connecting two general points of $C^{(k)}$) is strictly larger than the covering gonality.