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_{Y}X, corresponding to the identity X -> X,
and the `quasi-symmetric blow-up', Bl_{Y}X, 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.