Brasselet number and Newton polygons

TL;DR

This paper introduces a formula for computing the Brasselet number in non-degenerate complete intersections within toric varieties, and explores its invariance properties and implications for Euler obstruction in families of singularities.

Contribution

It provides a new formula for the Brasselet number and establishes invariance results for families of non-degenerate complete intersections, extending understanding of singularity invariants.

Findings

Derived a formula for the Brasselet number in toric varieties.

Proved invariance of the Brasselet number in certain families.

Established conditions for invariance of Euler obstruction in families with isolated singularities.

Abstract

We present a formula to compute the Brasselet number of $f:(Y,0)\to (\C, 0)$ where $Y\subset X$ is a non-degenerate complete intersection in a toric variety $X$. As applications we establish several results concerning invariance of the Brasselet number for families of non-degenerate complete intersections. Moreover, when $(X,0) = (\mathbb{C}^n,0)$ we derive sufficient conditions to obtain the invariance of the Euler obstruction for families of complete intersections with an isolated singularity which are contained in $X$.