On the index of contact and multiplicities for bigraded rings (Q5955029)

Let \(N\) be an \(n\)-dimensional complex manifold and let \(S\) (resp. \(Z\)) be a closed submanifold of \(N\) (resp. an analytic set in \(N\)). The main result of this paper is that the index of contact \(p(Z,S)(c)\) of \(Z\) with \(S\) at a point \(c \in \text{reg}(Z\cap S)\) at which \(Z\) is normal pseudo-flat along \(Z\cap S\) and the index of intersection \(g(Z,S)(c)\) of \(Z\) with \(S\) at the point \(c\) are equal. This implies that \(p(Z,S)(c)\) coincides with the Samuel multiplicity of the associated graded ring. The author also gives various methods of computing this invariant at such points.