A criterion for intersection multiplicity one

Abstract

Let X and Y be any pure dimensional subschemes of P n κ over an algebraically closed field K and let I{X) and I(Y) be the largest homogeneous ideals in K[x Q , , x n ] defining X and Y, respectively. By a pure dimensional subscheme X of P n κ 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 bounding the Stückrad-Vogel multiplicity

“…We give a quick precis of some of the main features of Stiickrad and Vogel's theory and of those properties of it which are of use here, borrowing heavily from [2] to do so.…”

“…The purpose of this note is to extend their main result, to set it in a wider context and to make some additional remarks on some of the arguments in [2].…”

“…See [2] for further background on such a criterion and for a sketch of this intersection theory; a fuller wellmotivated account is given in [16] (also, cf. [13]).…”

“…In [2], Achilles, Huneke and Vogel investigated the situation where the Stiickrad-Vogel multiplicity takes the value one. The purpose of this note is to extend their main result, to set it in a wider context and to make some additional remarks on some of the arguments in [2].…”

“…In a parallel development, Achilles, Huneke and Vogel [2] gave a criterion for intersection multiplicity one in the case where the component C is not necessarily proper, using the intersection theory of Stiickrad and Vogel. See [2] for further background on such a criterion and for a sketch of this intersection theory; a fuller wellmotivated account is given in [16] (also, cf. [13]).…”