Shadows of Blow-up Algebras

We study di erent 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, corresponding to the identity, and the `quasi-symmetric blow-up', corresponding to the embedding of X into a nonsingular variety. 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 is 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 rst carries the essential information computing Chern-Mather classes, and the second is likewise a carrier for Chern-SchwartzMacPherson 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 Fulton, and by Fulton and Johnson. We also identify a condition on the singularities of a hypersurface under which the quasi-symmetric blow-up is simply the linear ber space associated with a coherent sheaf.