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Summary: Proceedings of the Edinburgh Mathematical Society Submitted Paper
Paper 3 April 2003
SELF-INTERSECTIONS OF SURFACES AND WHITNEY
STRATIFICATIONS
R
UDIGER ACHILLES AND MIRELLA MANARESI
Dipartimento di Matematica, Universita 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 C n or P n and let CX (X X) be the normal cone to X in X X
(diagonally embedded). For a point x 2 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 Stuckrad{Vogel
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 strati cation of X by the multiplicity g(x) is a Whitney strati cation, the canonical
one if n = 3. The corresponding result for hypersurfaces in A n or P n , 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 strati cation.
AMS 2000 Mathematics subject classi cation: Primary 32S15
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