Euler characteristics of general linear sections and polynomial Chern classes

Paolo Aluffi

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 Zbb[t].