Point Configurations and Cayley-Menger Varieties

TL;DR

This paper explores the mathematical structures of point configurations across different spaces, linking them to algebraic varieties and demonstrating applications in geometry, mechanics, and theoretical physics.

Contribution

It establishes connections between point configuration classes and Cayley-Menger varieties, extending their relevance to various geometric and physical contexts.

Findings

Cayley-Menger varieties relate to Euclidean, Hermitian, and quaternionic configurations.

Applications include bounds for planar graph realizations and examples in Calabi-Yau manifolds.

The work bridges algebraic geometry with geometric and physical problems.

Abstract

Equivalence classes of $n$-point configurations in Euclidean, Hermitian, and quaternionic spaces are related, respectively, to classical determinantal varieties of symmetric, general, and skew-symmetric bilinear forms. Cayley-Menger varieties arise in the Euclidean case, and have relevance for mechanical linkages, polygon spaces and rigidity theory. Applications include upper bounds for realizations of planar Laman graphs with prescribed edge-lengths and examples of special Lagrangians in Calabi-Yau manifolds.