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arXiv · 1207.6638

Euler characteristics of general linear sections and polynomial Chern classes

Paolo Aluffi
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Abstract

We obtain a precise relation between the Chern-Schwartz-MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of formulas of Dimca-Papadima and Huh for the degrees of the polar map of a homogeneous polynomial, extending these formula to any algebraically closed field of characteristic~0, and proving a conjecture of Dolgachev on 'homaloidal' polynomials in the same context. We generalize these formulas to subschemes of higher codimension in projective space. We also describe a simple approach to a theory of `polynomial Chern classes' for varieties endowed with a morphism to projective space, recovering properties analogous to the Deligne-Grothendieck axioms from basic properties of the Euler characteristic. We prove that the polynomial Chern class defines homomorphisms from suitable relative Grothendieck rings of varieties to Z[t].

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Paolo Aluffi. 2012-07-27. Euler characteristics of general linear sections and polynomial Chern classes. https://doi.org/10.1007/s12215-013-0106-x

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