Mathematics – Algebraic Geometry
Scientific paper
2010-09-13
Mathematics
Algebraic Geometry
Minor revision. To appear in Arkiv f\"or Matematik
Scientific paper
10.1007/s11512-011-0164-2
Let X in V be a closed embedding, with V - X nonsingular. We define a constructible function on X, agreeing with Verdier's specialization of the constant function 1 when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence on the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich et al. The main property of the specialization function is a compatibility with the specialization of the Chern class of the complement V-X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier's result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart in a motivic group. The specialization function and the corresponding Chern class and motivic aspect all have natural `monodromy' decompositions, for for any X in V as above. The definition also yields an expression for Kai Behrend's constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.
Aluffi Paolo
No associations
LandOfFree
Verdier specialization via weak factorization does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Verdier specialization via weak factorization, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Verdier specialization via weak factorization will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-338284