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Mathematics > Algebraic Geometry

arXiv:1805.11116 (math)
[Submitted on 28 May 2018 (v1), last revised 20 Aug 2018 (this version, v2)]

Title:The Chern-Schwartz-MacPherson class of an embeddable scheme

Authors:Paolo Aluffi
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Abstract:There is an explicit formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety (in characteristic $0$) in terms of the Segre class of its jacobian subscheme; this has been known for a number of years. We generalize this formula to arbitrary embeddable schemes: for every subscheme $X$ of a nonsingular variety $V$, we define an associated subscheme $Y$ of a projective bundle over $V$ and provide an explicit formula for the Chern-Schwartz-MacPherson class of $X$ in terms of the Segre class of $Y$. If $X$ is a local complete intersection, a version of the result yields a direct expression for the Milnor class of $X$.
Comments: v2: References added, included a section on the relation with the Segre zeta function
Subjects: Algebraic Geometry (math.AG)
Cite as: arXiv:1805.11116 [math.AG]
  (or arXiv:1805.11116v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1805.11116
Journal reference: Forum of Mathematics, Sigma (2019), Vol. 7, e30, 28 pages
Related DOI: https://doi.org/10.1017/fms.2019.25

Submission history

From: Paolo Aluffi [view email]
[v1] Mon, 28 May 2018 18:06:51 UTC (17 KB)
[v2] Mon, 20 Aug 2018 22:54:02 UTC (21 KB)
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