Tohoku Mathematical Journal
- Tohoku Math. J. (2)
- Volume 56, Number 4 (2004), 593-619.
Shadows of blow-up algebras
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.
Tohoku Math. J. (2), Volume 56, Number 4 (2004), 593-619.
First available in Project Euclid: 11 April 2005
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Aluffi, Paolo. Shadows of blow-up algebras. Tohoku Math. J. (2) 56 (2004), no. 4, 593--619. doi:10.2748/tmj/1113246753. https://projecteuclid.org/euclid.tmj/1113246753