THE CHERN–SCHWARTZ–MACPHERSON CLASS OF AN EMBEDDABLE SCHEME
Forum of Mathematics, Sigma, Tome 7 (2019)
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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 $\mathscr{Y}$.
@article{10_1017_fms_2019_25,
author = {PAOLO ALUFFI},
title = {THE {CHERN{\textendash}SCHWARTZ{\textendash}MACPHERSON} {CLASS} {OF} {AN} {EMBEDDABLE} {SCHEME}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {7},
year = {2019},
doi = {10.1017/fms.2019.25},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.25/}
}
PAOLO ALUFFI. THE CHERN–SCHWARTZ–MACPHERSON CLASS OF AN EMBEDDABLE SCHEME. Forum of Mathematics, Sigma, Tome 7 (2019). doi: 10.1017/fms.2019.25
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