  I use Bridgeland's definition of a stability condition on a triangulated
category to investigate the stability of D-branes on Calabi-Yau cones given by
the canonical line bundle over a del Pezzo surface. In this context, I prove
the existence of the decay of a D3-brane into a set of fractional branes. This
is an important aspect of the derivation of quiver gauge theories from branes
at singularities via the technique of equivalences of categories. Some
important technical aspects of this equivalence are discussed. I also prove
that the representations corresponding to skyscraper sheaves supported off the
zero section are simple.
