Derived categories of (nested) Hilbert schemes

In this paper we provide several results regarding the structure of derived categories of (nested) Hilbert schemes of points. We show that the criteria of Krug-Sosna and Addington for the universal ideal sheaf functor to be fully faithful resp. a $\mathbb{P}$-functor are sharp. Then we show how to embed multiple copies of the derived category of the surface using these fully faithful functors. We also give a semiorthogonal decomposition for the nested Hilbert scheme of points on a surface, and finally we give an elementary proof of a semiorthogonal decomposition due to Toda for the symmetric product of a curve.