Local Euler obstructions of toric varieties

TL;DR

This paper develops algorithms to compute local Euler obstructions and dual degrees of toric varieties, applying them to weighted projective spaces and providing counterexamples to existing conjectures.

Contribution

It introduces new algorithms based on Matsui and Takeuchi's formula for toric A-discriminants, advancing the computational understanding of local Euler obstructions.

Findings

Counterexamples to Matsui and Takeuchi's conjecture

Only cones are defective normal toric surfaces

Algorithms for computing local Euler obstructions and dual degrees

Abstract

We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.