On Segre Numbers of Homogeneous Map Germs

Abstract

Segre numbers and Segre cycles of ideals were independently introduced by Tworzewski, by Achilles and Manaresi and by Gaffney and Gassler. They are generalization of the Lê numbers and Lê cycles, introduced by Massey. In this article we give Lê-Iomdine type formulas for these cycles and numbers of arbitrary ideals. As a consequence we give a Plücker type formula for the Segre numbers of ideals generated by weighted homogeneous functions, in terms of their weights and degree. As an application of these results, we compute, in a purely combinatorial manner, the Segre numbers of the ideal which defines the critical loci of a map germ defined by a sequence of central hyperplane arrangements in $\mathsf{C}^{n+1}$.