Abstract: Intersection cohomology, invented by Goresky and MacPherson, is a notion of cohomology for singular spaces which admits generalizations of classical theorems such as Poincare duality and the Lefschetz hyperplane theorem. It is constructed by taking global sections of a certain perverse sheaf called the intersection cohomology complex. This complex is itself an interesting topological invariant, and to study it one often looks at its characteristic cycle. In particular, if X is the affine cone over a projective variety Y, one can look at the multiplicity of this cycle over the vertex of X. I will discuss a conjecture of mine which would describe how this multiplicity changes with the projective embedding of Y, along with some evidence for the conjecture being true coming from the normal toric case.