Moving Curves of Least Gonality on Symmetric Products of Curves

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