The Chern-Schwartz-MacPherson class of an embeddable scheme
The Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety may be computed directly from the Segre class of the jacobian subscheme of the hypersurface; this has been known for a number of years. We generalize this fact to arbitrary embeddable schemes: for every subscheme X of a nonsingular variety V, we define an associated subscheme \cY of a projective bundle over V and provide an explicit formula for the Chern-Schwartz-MacPherson class of X in terms of the Segre class of \cY. If X is a local complete intersection, a version of the result yields a direct expression for the Milnor class of X. For V=P^n, we also obtain expressions for the Chern-Schwartz-MacPherson class of X in terms of the `Segre zeta function' of \cY.