Theo Raedschelders and I just uploaded our newest preprint: Noncommutative quadrics and Hilbert schemes of points. The main goal of the paper is to extend the result of Orlov's paper Geometric realizations of quiver algebras: we prove that (the derived category of) a noncommutative quadric embeds in (the derived category of) a commutative deformation of the Hilbert scheme of two points on a quadric. Orlov showed this for noncommutative planes (i.e. quadratic Artin--Schelter regular algebras), we now do the cubic Artin--Schelter regular case.

We also raise a question regarding the infinitesimal version of the picture above, whose statement works in smooth families. Namely, their common starting point is a fully faithful embedding $\mathbf{D}^{\mathrm{b}}(S)\hookrightarrow\mathbf{D}^{\mathrm{b}}(\mathrm{Hilb}^nS)$ for a sufficiently nice smooth projective surface. In this special case (I blogged before about lack of functoriality in general) we can induce a morphism $\mathrm{HH}^n(\mathrm{Hilb}^nS)\to\mathrm{HH}^n(S)$ , and we have an Hochschild–Kostant–Rosenberg for both sides, hence a block decomposition for these linear maps.

The question is now easy: in which cases do we have a surjection (or even isomorphism) $$\mathrm{H}^1(\mathrm{Hilb}^nS,\mathrm{T}_{\mathrm{Hilb}^nS})\twoheadrightarrow\mathrm{H}^0(S,\bigwedge^2\mathrm{T}_{S})?$$

This would mean that we can lift every infinitesimal noncommutative deformation of $S$ to a commutative deformation (as in Kodaira–Spencer) of the Hilbert scheme (ignoring obstructions). If this were indeed the case, it would be interesting to see whether it is possible to generalize Toda's result, showing that fully faithful functors (and not just equivalences) lift to deformations.

I have some work in progress addressing these questions, also for other moduli spaces (i.e. not just Hilbert schemes of points on surfaces). Stay tuned for more, and do not hesitate to ask questions.