SELF-INTERSECTIONS OF SURFACES AND WHITNEY STRATIFICATIONS

Abstract

Let $X$ be a surface in $\mathbb{C}^n$ or $\mathbb{P}^n$ and let $C_{X}(X\times X)$ be the normal cone to $X$ in $X\times X$ (diagonally embedded). For a point $x\in X$, denote by $g(x):=e_x(C_X(X\times X))$ the multiplicity of $C_X(X\times X)$ at $x$. It is a former result of the authors that $g(x)$ is the degree at $x$ of the Stückrad–Vogel cycle $v(X,X)=\sum_C j(X,X;C)[C]$ of the self-intersection of $X$, that is, $g(x)=\sum_Cj(X,X;C)e_x(C)$. 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 $\mathbb{A}^n$ or $\mathbb{P}^n$, diagonally embedded in a multiple product with itself, was conjectured by van Gastel. This is also discussed, but remains open.

AMS 2000 Mathematics subject classification: Primary 32S15. Secondary 13H15;14C17