This is a report on joint work in collaboration with A. Knutsen (University of Bergen, Norway).

The pairs $(S,H)$ with $S$ a complex K3 surface (i.e. $S$ is simply connected and ${\Omega }_S^2$ is trivial), and $H$ is an ample line bundle of genus $p$ (i.e. $H^2=2p-2$), have an irreducible moduli space ${𝒦}_p$ of dimension 19. Let $(S,H)\in {𝒦}_p$ be general. The linear system $\mid H\mid$ has dimension $p$ and its general element is a smooth curve of genus $p$. I consider $V_{\mid H\mid ,\delta }$, the Severi variety of $\delta$-nodal curves in $\mid H\mid$, which is locally closed of codimension $\delta$ in $\mid H\mid$ and its general element is an irreducible curve with exactly $\delta$ nodes and no other singularities, so that its geometric genus (i.e. the genus of its desingularization) is $g=p-\delta$. The image of $V_{\mid H\mid ,\delta }$ in ${ℳ}_g$, the moduli space of curves of genus $g$, has dimension $g$ and its importance in moduli problems is well known. For any integer $k\ge 2$, consider the subscheme $V_{\mid H\mid ,\delta }^k\subset V_{\mid H\mid ,\delta }$ given by $$\{C\in V_{\mid H\mid ,\delta }:\text{the normalization of}C\text{is a}k\text{-tuple cover of}{\mathbb{P}}^1\}.$$

The first question is: when is $V_k\subset V_{\mid H\mid ,\delta }$ not empty? Using a (by now classical) vector bundle technique due to Lazarsfeld, we show that a necessary condition is $\delta \ge \alpha (g-(k-1)(\alpha +1))$ where $\alpha :=\lfloor \frac{g}{2(k-1)}\rfloor$. A dimension count shows that, if $V_{\mid H\mid ,\delta }^k$ is not empty, its expected dimension should be $2k-2$. As a matter of fact, we prove that for all even $p$ and all $\delta ,k$ verifying the above relation, $V_{\mid H\mid ,\delta }^k$ is actually not empty, of dimension $2k-2$. For $p$ odd we have a slightly weaker result. Finally, I will talk about some interesting relations of these results with a conjecture by Hassett and Tschinkel on the Mori cone of the hyperkhähler manifold ${\text{Hilb}}^k(S)$ (which parametrizes subschemes of $S$ of finite lenght $k$).