Bi-Lipschitz equivalent cones with different degrees

TL;DR

The paper constructs complex algebraic cones that are bi-Lipschitz equivalent but have different degrees, and explores properties of projective hypersurfaces and links of real cones.

Contribution

It demonstrates the existence of bi-Lipschitz equivalent cones with different degrees and classifies certain real cone links, advancing understanding of algebraic and geometric equivalences.

Findings

Existence of bi-Lipschitz equivalent cones with different degrees

Homeomorphic projective hypersurfaces have the same degree

Classification of links of real cones with specific bases

Abstract

We show that for every $k\ge 3$ there exist complex algebraic cones of dimension $k$ with isolated singularities, which are bi-Lipschitz and semi-algebraically equivalent but they have different degrees. We also prove that homeomorphic projective hypersurfaces with dimension greater than 2 have the same degree. In the final part of the paper, we classify links of real cones with base $\mathbb{P}^1\times \mathbb{P}^2.$ As an application we give an example of three four dimensional real algebraic cones in $\mathbb{R}^8$ with isolated singularity which are semi-algebraically and bi-Lipschitz equivalent but they have non-homeomorphic bases.