Euclidean distance degree of projective varieties

TL;DR

This paper confirms a conjecture linking the Euclidean distance degree of singular projective varieties to the local Euler obstruction function, providing a new way to compute this geometric measure.

Contribution

It proves a conjecture that relates the Euclidean distance degree to the local Euler obstruction, extending the understanding to possibly singular varieties.

Findings

Confirmed the conjecture for singular projective varieties.

Established a formula connecting Euclidean distance degree with local Euler obstruction.

Enhanced methods for computing Euclidean distance degrees in algebraic geometry.

Abstract

We give a positive answer to a conjecture of Aluffi-Harris on the computation of the Euclidean distance degree of a possibly singular projective variety in terms of the local Euler obstruction function.