Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's [paper](http://arxiv.org/abs/math/0503632))
as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full subcategory of perfect complexes.


For example, there is a quiver description in the case of ADE surface singularities:
<http://arxiv.org/abs/math/0511155>


Are there any other cases do we have results for the category of singularities? In particular, for higher codimension varieties?



For what varieties do we have results on the category of singularities?
A student of Orlov is working on one case. He will give a talk in two weeks in [Padova](http://events.math.unipd.it/hsaa/?q=node/5#), here is the abstract:


* Oleksandr Kravets (Moscow Higher School of Economics, Russia), *Exceptional collections in categories of singularities of 3-dimensional Landau-Ginzburg models*.


Abstract:
Derived categories of singularities measure how far is an algebraic variety
from being smooth. They appear on the algebraic side of the Homological
Mirror Symmetry (HMS) for the Landau-Ginzburg models. The talk will be
dedicated to the case when the superpotential is given by the so called
invertible polynomial. In the case of small dimensions, I will give
the description of mentioned categories in terms of full strongly exceptional collections
of good kind. I will also recall several conjectures concerning HMS for the
current case and will show how the constructed collections help to assert


