Point Configurations and Cayley-Menger Varieties
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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.
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Terracini matroids: algebraic matroids of secants and embedded joins
Defines Terracini unions of matroids that capture when the algebraic matroid of a join coincides with the matroid union of its summands, motivated by Terracini's lemma, with applications to toric surfaces and threefolds.
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