Intersection multiplicities and tangent cones
B. R. Tennison · Mathematical Proceedings of the Cambridge Philosophical Society · 1979
The following result has at least the appeal of intuitive plausibility. Let U and V be subvarieties of an algebraic variety X; let x ∈ X be an isolated point of the intersection of U and V, and let I(X, U. V, x) denote the intersection multiplicity (in some sense to be made precise) of U and V at x. Now let Tx be the tangent space to X at x, within which lie the tangent cones Cu, Cv to U and V at x. Then