TITLE: Hilbert schemes of points and exceptional collections
QUESTION [7 upvotes]: Does there exist an exceptional collection of coherent sheaves on the Hilbert scheme of points on projective plane? If so, Could it be a strong full collection?

REPLY [9 votes]: It's more than 7 years after the question, but better late than never. The result of @Sasha holds for any number of points, and for surfaces more general than $\mathbb{P}^2$: once $\mathbf{D}^{\mathrm{b}}(S)$ has a full exceptional collection, then so does $\mathbf{D}^{\mathrm{b}}(\mathop{\mathrm{Hilb}}^nS)$. This is proposition 1.3 of
Krug, Andreas; Sosna, Pawel, On the derived category of the Hilbert scheme of points on an Enriques surface, Sel. Math., New Ser. 21, No. 4, 1339-1360 (2015). ZBL1331.18015.