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Proceedings of the Edinburgh Mathematical Society Submitted Paper Paper 3 April 2003
 

Summary: Proceedings of the Edinburgh Mathematical Society Submitted Paper
Paper 3 April 2003
SELF-INTERSECTIONS OF SURFACES AND WHITNEY
STRATIFICATIONS
RšUDIGER ACHILLES AND MIRELLA MANARESI
Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta S. Donato 5,
I-40126 Bologna, Italy (achilles@dm.unibo.it, manaresi@dm.unibo.it)
(Received )
Abstract Let X be a surface in Cn or Pn and let CX (X × X) be the normal cone to X in X × X
(diagonally embedded). For a point x X, denote by g(x) := ex(CX (X × X)) the multiplicity of
CX (X × X) at x. It is a former result of the authors that g(x) is the degree at x of the Stšuckrad­Vogel
cycle v(X, X) =
P
C j(X, X; C) [C] of the self-intersection of X, that is, g(x) =
P
C j(X, X; C) ex(C).
We prove that the stratification of X by the multiplicity g(x) is a Whitney stratification, the canonical
one if n = 3. The corresponding result for hypersurfaces in An or Pn, diagonally embedded in a multiple
product with itself, was conjectured by L. van Gastel. This is also discussed, but remains open.
Keywords: Hypersurface singularities, normal cone, Whitney stratification.

  

Source: Achilles, Rüdiger - Dipartimento di Matematica, Università di Bologna

 

Collections: Mathematics