Shadows of blow-up algebras

Paolo Aluffi

We study different notions of blow-up of a scheme X along a subscheme Y, depending on the datum of an embedding of X into an ambient scheme. The two extremes in this theory are the ordinary blow-up, Bl_YX, corresponding to the identity X -> X, and the `quasi-symmetric blow-up', Bl_YX, corresponding to an embedding X -> M into a nonsingular variety M. We prove that this latter blow-up is intrinsic of Y and X, and is universal with respect to the requirement of being embedded as a subscheme of the ordinary blow-up of some ambient space along Y.

We consider these notions in the context of the theory of characteristic classes of singular varieties. We prove that if X a hypersurface in a nonsingular variety and Y is its `singularity subscheme', these two extremes embody respectively the conormal and characteristic cycles of X. Consequently, the first carries the essential information computing Chern-Mather classes, and the second is likewise a carrier for Chern-Schwartz-MacPherson classes. In our approach, these classes are obtained from Segre class-like invariants, in precisely the same way as other intrinsic characteristic classes such as those proposed by William Fulton, and by W. Fulton and Kent Johnson.

We also identify a condition on the singularities of a hypersurface under which the quasi-symmetric blow-up is simply the linear fiber space associated with a coherent sheaf.