Rüdiger Achilles

Intersection numbers, Segre numbers and generalized Samuel multiplicities

Abstract. We prove that the analytic intersection numbers (extended index of intersection) of Tworzewski and the Segre numbers of Gaffney and Gassler are generalized Samuel multiplicities, which have been introduced for an arbitrary ideal in an arbitrary Noetherian local ring by Manaresi and the first author.

A criterion for intersection multiplicity one

Let X and Y be any pure dimensional subschemes of P n k over an algebraically closed field K and let I(X ) and I(Y ) be the largest homogeneous ideals in K [x 0 ,…, x n ] defining X and Y , respectively. By a pure dimensional subscheme X of P n k we shall always mean a closed pure dimensional subscheme without imbedded components, i.e., all primes belonging to I ( X ) have the same dimension.

On the self-intersection cycle of surfaces and some classical formulas for their secant varieties

Abstract We study the relation between certain local and global numerical invariants of a projective surface given by the Stückrad–Vogel self-intersection cycle. In the smooth case we find a relation between the degrees of the tangent, the secant and the first polar variety which generalizes classical relations between the so-called elementary invariants of surfaces. For singular surfaces this relation involves also the contributions of singularities to the self-intersection cycle. In order to make the formula really effective for singular surfaces, we need to describe the movable part of the self-intersection cycle on the singular locus. This is done explicitly for a particular class of non-normal del Pezzo surfaces.

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.