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arXiv · 1604.07954

Towards a simple characterization of the Chern-Schwartz-MacPherson class

James Fullwood·Dongxu Wang
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Abstract

For a large class of possibly singular complete intersections we prove a formula for their Chern-Schwartz-MacPherson classes in terms of a single blowup along a scheme supported on the singular loci of such varieties. In the hypersurface case our formula recovers a formula of Aluffi proven in 1996. As our formula is in no way tailored to the complete intersection hypothesis, we conjecture that it holds for all closed subschemes of a smooth variety. If in fact true, such a formula would provide a simple characterization of the Chern-Schwartz-MacPherson class which does not depend on a resolution of singularities. We also show that our formula may be suitably interpreted as the Chern-Fulton class of a scheme-like object which we refer to as an `$\mathfrak{f}$-scheme'.

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James Fullwood, Dongxu Wang. 2016-04-27. Towards a simple characterization of the Chern-Schwartz-MacPherson class. https://arxiv.org/abs/1604.07954

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