THE CHERN–SCHWARTZ–MACPHERSON CLASS OF AN EMBEDDABLE SCHEME

Abstract

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$\mathscr{Y}$.