Bi-Lipschitz equivalent cones with different degrees

Author:

Alexandre Fernandes, Zbigniew Jelonek, José Edson Sampaio

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Complex Variables (math.CV), Metric Geometry (math.MG)

journal:

date:

2023-09-12 16:00:00

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.

PDF: Bi-Lipschitz equivalent cones with different degrees.pdf