SELF-INTERSECTIONS OF SURFACES AND WHITNEY STRATIFICATIONS

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) = 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 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.