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.
Dutta et al. (Fri,) studied this question.
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