Multitoric surfaces and Euler obstruction of a function

Abstract

In this work, we present a formula to compute the Euler obstruction of a function [Formula: see text] and its Brasselet number, where [Formula: see text] is a multitoric surface. As an application of this formula, we compute the Euler obstruction of a function on some families of determinantal surfaces.

Local topology of a deformation of a function-germ with a one-dimensional critical set

“…In [6], the authors proved several formulas about the local topology of the generalized Milnor fibre of a function germ f using the Brasselet number, like the Lê-Greuel type formula (Theorem 4.2 in [6]): B f,X (0) − B f,X g (0) = (−1) dim C n, where n is the number of stratified Morse critical points of a Morsefication of g| X∩f −1 (δ)∩Bǫ on V q ∩f −1 (δ)∩B ǫ . In [5], Dalbelo e Pereira provided formulas to compute the Brasselet number of a function defined over a toric variety and in [1], Ament, Nuño-Ballesteros, Oréfice-Okamoto and Tomazella computed the Brasselet number of a function-germ with isolated singularity at the origin and defined over an isolated determinantal variety (IDS) and the Brasselet number of finite functions defined over a reduced curve. More recently, in [4], Dalbelo and Hartmann calculated the Brasselet number of a function-germ defined over a toric variety using combinatorical properties of the Newton polygons.…”