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 $\mathscr{Y}$ of a projective bundle $\mathscr{V}$ over $V$ and provide an explicit formula for the Chern–Schwartz–MacPherson class of $X$ in terms of the Segre class of $\mathscr{Y}$ in $\mathscr{V}$ . If $X$ is a local complete intersection, a version of the result yields a direct expression for the Milnor class of $X$ . For $V=\mathbb{P}^{n}$ , we also obtain expressions for the Chern–Schwartz–MacPherson class of $X$ in terms of the ‘Segre zeta function’ of $\mathscr{Y}$ .