Moving Curves of Least Gonality on Symmetric Products of Curves

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.