  For a large class of possibly singular complete intersections we prove a
formula for their Chern-Schwartz-MacPherson classes in terms of a single blowup
along a scheme supported on the singular loci of such varieties. In the
hypersurface case our formula recovers a formula of Aluffi proven in 1996. As
our formula is in no way tailored to the complete intersection hypothesis, we
conjecture that it holds for all closed subschemes of a smooth variety. If in
fact true, such a formula would provide a simple characterization of the
Chern-Schwartz-MacPherson class which does not depend on a resolution of
singularities. We also show that our formula may be suitably interpreted as the
Chern-Fulton class of a scheme-like object which we refer to as an
`$\mathfrak{f}$-scheme'.
