Moving curves of least gonality on symmetric products of curves (Q6632419)

For any irreducible complex projective curve \(E\) its gonality is the the gonality of its normalization \(F\), i.e. the minimum degree of a surjective morphism \(F\to \mathbb{P^1}\). For any irreducible complex projective variety \(Y\) its covering gonality \(\mathrm{cov.gon}(Y)\) is the minimal gonality of an irreducible curve containing the general point of \(Y\). The connecting gonality \(\mathrm{conn.gon}(Y)\) of \(Y\) is the minimal gonality of a curve \(E\subset Y\) passing through \(2\) general points of \(Y\). Fix \(k\in \{2,3,4\}\). For any smooth curve \(C\) of genus \(g\) let \(C^{(k)}\) be the symmetric product of \(k\) copies of \(C\). In previous papers the authors proved that for all \(g\ge k+4\) we have \(\mathrm{gon}(C) =\mathrm{cov.gon}(C^{(k)})\). In the present paper (assuming \(C\) neither hyperelliptic nor bielliptic) the authors describe all the families evincing \(\mathrm{cov.gon}(C^{(k)})\). As a corollary they prove that \(\mathrm{conn.gon}(C^{(k)})>\mathrm{cov.gon}(C^{(k)})\)