Shadows of blow-up algebras

Abstract

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, 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 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 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 fiber space associated with a coherent sheaf.