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 an embedding 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 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 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.

23 pages. Substantial revision of the January 2002 version

References in corpus (6)

• Characteristic Classes of Hypersurfaces and Characteristic Cycles
• Chern classes for singular hypersurfaces
• Computing characteristic classes of projective schemes
• A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes
• Weighted Chern-Mather classes and Milnor classes of hypersurfaces
• Interpolation of characteristic classes of singular hypersurfaces

Cited by in corpus (10)

• Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
• An algorithm for computing the topological Euler characteristic of complex projective varieties
• Segre classes and invariants of singular varieties
• Local Euler Obstruction and Chern-Mather classes of Determinantal Varieties
• The Segre zeta function of an ideal
• Involutions of varieties and Rost's degree formula
• Geometric Invariants of Recursive Group Orbit Stratification
• Chern Classes of Logarithmic Derivations for Free Divisors with Jacobian Ideal of Linear Type
• The Valabrega-Valla modules of monomial ideals
• The Aluffi Algebra of a hypersurface with isolated singularity