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THE CHERN–SCHWARTZ–MACPHERSON CLASS OF AN EMBEDDABLE SCHEME

Part of: Surfaces and higher-dimensional varieties Cycles and subschemes

Published online by Cambridge University Press:  18 September 2019

PAOLO ALUFFI
PAOLO ALUFFI*
Affiliation:
Mathematics Department, Florida State University, Tallahassee FL 32306, USA; aluffi@math.fsu.edu

Abstract

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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 $\mathscr{Y}$.

MSC classification

Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities
Secondary: 14J17: Singularities
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e30
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2019

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