Bi-Lipschitz equivalent cones with different degrees

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.

13 pages. arXiv admin note: text overlap with arXiv:2302.05387