For a nonsingular variety sitting inside a nonsingular ambient variety there is a semi-orthogonal decomposition of the derived category of the blow-up (with center that subvariety).
What can be said about singular varieties? I am mostly interested in the case where the ambient variety is singular but the subvariety to be blown up is nonsingular. It would also be nice to know results about both the bounded derived category and the category of perfect complexes.



Is there a blow-up formula for the derived category of a singular ambient variety?
In the simplest example of a 2-dimensional quadratic cone, the derived category of the blowup is indecomposable over the base. On the other hand, in this paper <http://link.springer.com/article/10.1007/s00029-008-0052-1> there are examples of interesting semiorthogonal decompositions of such blowups.


