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$\mathscr{Y}$.
Highlights
The goal of this paper is the generalization to arbitrary subschemes of nonsingular varieties of a twenty-year old formula for the Chern–Schwartz–MacPherson class of hypersurfaces, in terms of the Segre class of an associated scheme
In [Alu99], we proved a formula for the Chern–Schwartz–MacPherson class of a hypersurface in a nonsingular variety, in terms of the Segre class of its singularity subscheme. (In particular, this yields a formula for the topological Euler characteristic of arbitrary hypersurfaces of nonsingular varieties.) Applications include computations in enumerative geometry [Alu98], singularities of logarithmic foliations [CSV06], Sethi–Vafa–Witten-type formulas [AE09], and others
For ι : X → V an arbitrary closed embedding of a scheme X in a nonsingular variety V, we provide a formula for ι∗cSM(X ) ∈ A∗V in terms of the Segre class s(Y, V ) of an associated subscheme Y of a projective bundle V over V
Summary
The goal of this paper is the generalization to arbitrary subschemes of nonsingular varieties of a twenty-year old formula for the Chern–Schwartz–MacPherson class of hypersurfaces, in terms of the Segre class of an associated scheme. In [Alu99], we proved a formula for the Chern–Schwartz–MacPherson class of a hypersurface in a nonsingular variety, in terms of the Segre class of its singularity subscheme. In [CBMS], this hypersurface is constructed in the local complete intersection case, and it is used to obtain formulas for Milnor classes (see Section 3.6). It would be interesting to relate the center of the blow-up in [FW] to Y
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