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 $\mathscr{Y}$ .

Chern Classes of Splayed Intersections

We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a splayedness assumption. We show that the relation for the Chern–Schwartz–MacPherson classes holds for two splayed hypersurfaces in a nonsingular variety, and under a strong splayedness assumption for more general subschemes. Moreover, the relation is shown to hold for the Chern–Fulton classes of any two splayed subschemes. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.

Divisors on Varieties Over a Real Closed Field

Let X be a projective nonsingular variety over a real closed field R such that the set X(R) of R-rational points of X is nonempty. Let ClR(X) = Cl(X)/Γ(X), where Cl(X) is the group of classes of linearly equivalent divisors on X and Γ(X) is the subgroup of Cl(X) consisting of the classes of divisors whose restriction to some neighborhood of X(R) in X is linearly equivalent to 0. It is proved that the group ClR(X) is isomorphic to (Z/2)s for some non-negative integer s. Moreover, an upper bound on s is given in terms of the Z/2-dimension of the group cohomology modules of Gal(C/R), where , with values in the Néron-Severi group and the Picard variety of Xc = X xR C.