On Multiplicities of Non-isolated Intersection Components

In 1942, B.L. van der Waerden observed a very interesting fact in [24]. (We translate here from the original German): "However, to the best of my knowledge, the multiplicity of a non-isolated intersection point of 3 surfaces has never yet been defined." (For a wealth of background material see, e.g., [9, 10, 13, 21]. Of course, it is very difficult to investigate embedded primary components, see, e.g., [11].) The focus of this paper is a discussion of this observation. First, our theorem 1 shows that indeed we need contributions of non-isolated intersection components to the intersection theory in P as developed in [4] and [18, 20] (see [7,3] for the relation between [4] and [20].). Second, our theorem 2 establishes that non-isolated intersection points of 3 surfaces have not always multiplicities. Third, our theorem 3 shows that certain non-isolated intersection points do indeed have multiplicities. This is demonstrated by examples. Here is our first example (see also [19]).