The Bottleneck Degree of Algebraic Varieties
Sandra Di Rocco, David Eklund, Madeleine Weinstein · SIAM Journal on Applied Algebra and Geometry · 2020
A bottleneck of a smooth algebraic variety $X \subset \mathbb{C}^n$ is a pair $(x,y)$ of distinct points $x,y \in X$ such that the Euclidean normal spaces at $x$ and $y$ contain the line spanned by $x$ and $y$. The narrowness of bottlenecks is a fundamental complexity measure in the algebraic geometry of data. In this paper we study the number of bottlenecks of affine and projective varieties, which we call the bottleneck degree. The bottleneck degree is a measure of the complexity of computing all bottlenecks of an algebraic variety, using, for example, numerical homotopy methods. We show that the bottleneck degree is a function of classical invariants such as Chern classes and polar classes. We give the formula explicitly in low dimension and provide an algorithm to compute it in the general case.