Point Configurations and Cayley-Menger Varieties
نویسنده
چکیده
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 CalabiYau manifolds. Introduction. We are concerned, initially, with configurations of n labeled (or ordered) points in the Euclidean space R, up to equivalence under congruence and similarity (rescaling). We require at least two points to be distinct and denote by Cn(R ) the resulting configuration space, made of such equivalence classes. Cayley-Menger varieties appear when point configurations are looked upon as encoded in the information given by the squared distances between any pair of points (up to proportionality). Cayley expressed the necessary relations between these squared distances as the vanishing of certain determinants, and Menger found sufficient conditions for a set of solutions to actually represent the mutual squared distances of a point configuration. These conditions amount to sign requirements on determinants of the same kind [3] [4]. Here, we look at the matter not so much in terms of distance geometry, as envisaged e.g. in [4] [10], but rather in terms of algebraic geometry. In fact, our medium will be mostly that of complex algebraic-geometry, since we are also interested in certain complexifications of configuration spaces (cf. [5]). Thus, we define the Cayley-Menger variety CM(C) as the Zariski-closure of imCn(R ) ⊂ (2)−1 (C).
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تاریخ انتشار 2002