Self-Intersections of Surfaces and Whitney Stratifications

Abstract

Let X be a surface in C n or P n and let C X (X × X) be the normal cone to X in X × X (diagonally embedded). For a point x ∈ X, denote by g( is a former result of the authors that g(x) is the degree at x of the Stückrad-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 van Gastel. This is also discussed, but remains open.