MacPherson's And Fulton's Chern Classes Of Hypersurfaces<P>
Paolo Aluffi<P>

We prove that MacPherson's total Chern class of a singular
hypersurface agrees `numerically' with a class obtained by means of
Fulton's intrinsic class of a scheme.  More precisely, for a
hypersurface X with Jacobian scheme J, and any positive integer t,
consider the class P(X,J,t) obtained by taking Fulton's class of the
t-thickening of X along J. Then P(X,J,t) is a polynomial in t (with
coefficients in the Chow group of X), and we show that P(X,J,-1)
agrees with MacPherson's class of X after push-forward via the map
defined by the linear system of X.<P>

We conjecture the equality holds at the level of Chow groups, and
speculate that a similar result should hold for arbitrary algebraic
schemes in characteristic 0.