Summary: Proceedings of the Edinburgh Mathematical Society Submitted Paper
Paper 3 April 2003
SELF-INTERSECTIONS OF SURFACES AND WHITNEY
RšUDIGER ACHILLES AND MIRELLA MANARESI
Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta S. Donato 5,
I-40126 Bologna, Italy (firstname.lastname@example.org, email@example.com)
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) =
C j(X, X; C) [C] of the self-intersection of X, that is, g(x) =
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.