Published online by Cambridge University Press: 18 September 2019
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
$\mathscr{Y}$.