Nef cones of Hilbert schemes of points on some K3 surfaces

Let X be a Mori dream K3 surface of Picard rank $2$. We compute the nef cone of the Hilbert scheme X[n] of n points on X for large n using Bridgeland stability methods. We also find bounds on the nef cone for small n. Studying the geometry of a $2$-to-$1$ cover X→ Q of a smooth quadric Q in P³, and its induced rational map X[2] Q[2] on the Hilbert schemes, we calculate the nef cone of X[2] in a particular case. Finally, we also find the nef cone of the nested Hilbert scheme X[n,n+1] for large n.