The Bruce-Roberts Numbers of a Function on an ICIS

Abstract

We give formulas for the Bruce-Roberts number µ BR (f, X) and its relative version µ − BR (f, X) of a function f with respect to an ICIS (X, 0). We show that, where µ and τ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for µ BR (f, X) is more complicated and also involves µ(f ) and some lengths in terms of the ideals I X and Jf . We also consider the logarithmic characteristic variety, LC(X), and its relative version, LC(X) − . We show that LC(X) − is Cohen-Macaulay and that LC(X) is Cohen-Macaulay at any point not in X × {0}. We generalize previous results presented by the authors when (X, 0) has codimension one and by Bruce and Roberts when it is weighted homogeneous of any codimension.