Point Configurations and Cayley-Menger Varieties

2002/07/12 by Ciprian S. Borcea, Borcea, Ciprian S. · 5 citations
Computer Science · Mathematics · #14J32 #14M12 #14P25 #Advanced Graph Theory Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #math.AG #msc:14J32 #msc:14M12 #msc:14P25 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0207110

arxiv created 2002/07/12 · openalex publication_date 2002/07/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Terracini matroids: algebraic matroids of secants and embedded joins
• Mixed Volume Techniques for Embeddings of Laman Graphs
• Determining Generic Point Configurations From Unlabeled Path or Loop Lengths
• On tangents to quadric surfaces
• Algebraic methods for counting Euclidean embeddings of rigid graphs

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