On multiplicities for improper intersections.

*(English)*Zbl 0814.14045From the introduction:

In 1937, Zariski proved the following statement: If the origin is an \(m_ i\)-fold point of \(n\) hypersurfaces \(F_ 1,\dots, F_ n\) of \(\mathbb{P}^ n_ K\) and it is an isolated point of intersection of these \(n\) varieties, then the intersection multiplicity at the origin is not less than \(m_ 1 \cdot m_ 2 \cdot \cdots \cdot m_ n\) under the assumption that the hypersurfaces \(F_ 1, \dots, F_ n\) have only a finite number of common points. The case \(n = 3\) was studied by Berzolari in 1896. Nowadays such results are special cases in intersection theory for proper components. – An aim of this paper is to extend these investigations even to improper components (maybe embedded). Hence our main result is a variation of the classical fundamental result about positivity of intersection numbers of pure-dimensional schemes meeting in isolated points. Considering refined Bézout’s theorem we replace the product of the degrees of schemes by the product of their local multiplicities at an arbitrary subvariety of the intersection. Therefore this statement can be considered as a certain local version of the refined theorem of Bézout.

In 1937, Zariski proved the following statement: If the origin is an \(m_ i\)-fold point of \(n\) hypersurfaces \(F_ 1,\dots, F_ n\) of \(\mathbb{P}^ n_ K\) and it is an isolated point of intersection of these \(n\) varieties, then the intersection multiplicity at the origin is not less than \(m_ 1 \cdot m_ 2 \cdot \cdots \cdot m_ n\) under the assumption that the hypersurfaces \(F_ 1, \dots, F_ n\) have only a finite number of common points. The case \(n = 3\) was studied by Berzolari in 1896. Nowadays such results are special cases in intersection theory for proper components. – An aim of this paper is to extend these investigations even to improper components (maybe embedded). Hence our main result is a variation of the classical fundamental result about positivity of intersection numbers of pure-dimensional schemes meeting in isolated points. Considering refined Bézout’s theorem we replace the product of the degrees of schemes by the product of their local multiplicities at an arbitrary subvariety of the intersection. Therefore this statement can be considered as a certain local version of the refined theorem of Bézout.

Reviewer: E.Stagnaro (Padova)