We illustrate the typical usage of Bayer and Macr\`{ i}'s Positivity Lemma to compute the nef cones of the Hilbert schemes $X^{ [n]}$ by combining the Bridgeland stability methods (for large $n$) and classical methods (for small $n$). We use Mori dream K3 surfaces $X$ of Picard rank $2$ as our working example. We also compute the nef cones of the nested Hilbert schemes $X^{ [n,n+1]}$ for such $X$, for large $n$.
Categories: math.AG