Multiplicities of a bigraded ring and intersection theory
Let \(X\) and \(Y\) be projective varieties in \({\mathbb P}^n_K\). The Samuel multiplicity of an \(\mathfrak m\)-primary ideal in a local ring \((A,{\mathfrak m})\) can be used to define the intersection number of an irreducible component of the intersection of \(X\) and \(Y\). In this paper the authors define a multiplicity sequence \(c_0 (I,A),\dots,c_d (I,A)\) for an arbitrary ideal \(I\) of a \(d\)-dimensional local ring \((A,{\mathfrak m})\) which is closely related to the Stückrad-Vogel intersection cycle. It is defined by means of a certain bigraded ring \(G_{\mathfrak m} (G_I (A))\). The main result of this paper implies that each number of the multiplicity sequence equals the (local) degree of the part of the cycle in a certain dimension. Applications include an interpretation of the Segre classes of a subscheme as multiplicities in a bigraded ring, and a local version of Bezout's theorem. In the case where \(I\) is \(\mathfrak m\)-primary, \(c_0 (I,A)\) is the Samuel multiplicity of \(I\) and it is the only element of the sequence which is non-zero. If the embedded join of \(X\) and \(Y\) has minimal dimension, then again the sequence reduces to only one element different from zero. Interesting connections are made to earlier papers by the authors.
scientific article; zbMATH DE number 1121712