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Characterizing generic global rigidity
"... Abstract. A ddimensional framework is a graph and a map from its vertices to E d. Such a framework is globally rigid if it is the only framework in E d with the same graph and edge lengths, up to rigid motions. For which underlying graphs is a generic framework globally rigid? We answer this questi ..."
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Cited by 17 (1 self)
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Abstract. A ddimensional framework is a graph and a map from its vertices to E d. Such a framework is globally rigid if it is the only framework in E d with the same graph and edge lengths, up to rigid motions. For which underlying graphs is a generic framework globally rigid? We answer this question by proving a conjecture by Connelly, that his sufficient condition is also necessary: a generic framework is globally rigid if and only if it has a stress matrix with kernel of dimension d + 1, the minimum possible. An alternate version of the condition comes from considering the geometry of the lengthsquared mapping ℓ: the graph is generically locally rigid iff the rank of ℓ is maximal, and it is generically globally rigid iff the rank of the Gauss map on the image of ℓ is maximal. We also show that this condition is efficiently checkable with a randomized algorithm, and prove that if a graph is not generically globally rigid then it is flexible one dimension higher. 1.
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 15 (1 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Sextic double solids
, 2004
"... Abstract. We prove nonrationality and birational superrigidity of a Qfactorial double cover X of P 3 ramified along a sextic surface with at most simple double points. We also show that the condition #Sing(X)  ≤ 14 implies Qfactoriality of X. In particular, every double cover of P 3 with at m ..."
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Cited by 12 (5 self)
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Abstract. We prove nonrationality and birational superrigidity of a Qfactorial double cover X of P 3 ramified along a sextic surface with at most simple double points. We also show that the condition #Sing(X)  ≤ 14 implies Qfactoriality of X. In particular, every double cover of P 3 with at most 14 simple double points is nonrational and not birationally isomorphic to a conic bundle. All the birational transformations of X into elliptic fibrations and into Fano 3folds with canonical singularities are classified. We consider some relevant problems over fields of finite characteristic. When X is defined over a number field F we prove that the set of rational points on the 3fold X is potentially dense if Sing(X) ̸ = ∅.
Gimigliano, Secant varieties of Grassmann Varieties
 Proc. Am. Math. Soc
"... Abstract. We consider the dimensions of the higher secant varieties of the Grassmann varieties. We give new instances where these secant varieties have the expected dimension and also a new example where a higher secant variety does not. ..."
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Cited by 7 (0 self)
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Abstract. We consider the dimensions of the higher secant varieties of the Grassmann varieties. We give new instances where these secant varieties have the expected dimension and also a new example where a higher secant variety does not.
ON THE IRREDUCIBILITY OF SECANT CONES, AND AN APPLICATION TO LINEAR NORMALITY
"... Given a smooth subvariety of dimension greater than (2/3)(r − 1) in P r, we show that the double locus (upstairs) of its generic projection to P r−1 is irreducible. This implies a version of Zak’s linear normality theorem. A classical, and recently revisited (see [GP], [L], [Pi], and references ther ..."
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Given a smooth subvariety of dimension greater than (2/3)(r − 1) in P r, we show that the double locus (upstairs) of its generic projection to P r−1 is irreducible. This implies a version of Zak’s linear normality theorem. A classical, and recently revisited (see [GP], [L], [Pi], and references therein), method for studying the geometry of a subvariety Y ⊂ P r is to project Y generically to a lowerdimensional projective space, for example, so that Y maps birationally to a (singular) hypersurface ¯Y ⊂ P m+1. To make use of this method, it is usually important to have precise control over the singularities of ¯Y and in particular over the entire singular (i.e., double) locus DY of ¯Y and its inverse image CY in Y. As the dimension of these is easily determined, a natural question is the following: Are CY and DY irreducible? This question plays an important role, for instance, in H. Pinkham’s work on regularity bounds for surfaces [Pi]. The purpose of this paper is to show that this irreducibility holds provided the codimension of Y is sufficiently small compared to its dimension (see Theorems 1, 2 and Corollary 3). As an application we give a proof of Zak’s linear normality theorem (in a slightly restricted range; see Corollary 5). Indeed, the results seem closely related as our argument ultimately depends on having a bound on the dimension of singular loci of hyperplane sections, manifested in the form of the integer σ (Y) (see Theorem 1), and it is Zak’s theorem on tangencies—also a principal ingredient in other proofs of linear normality—that gives us good control over σ (Y). We begin with some definitions. Let Y denote an irreducible mdimensional subvariety of P r. As usual, we mean by a secant line of Y a limit of lines in P r spanned