MacPherson's And Fulton's Chern Classes Of Hypersurfaces
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.
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.