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31
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 40 (4 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.
Quantum cohomology of minuscule homogeneous spaces II Hidden symmetries
, 2008
"... We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points GromovWitten invariants can always be interpreted as classical intersection numbers on auxiliary homogeneous varieties. Our main combinatorial tools are certain quivers, in ..."
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Cited by 36 (15 self)
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We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points GromovWitten invariants can always be interpreted as classical intersection numbers on auxiliary homogeneous varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In particular we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties.
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 35 (5 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.
Differentialgeometric characterizations of complete intersections
 Jour. Diff. Geom
, 1996
"... Abstract. We characterize complete intersections in terms of local differential geometry. Let Xn ⊂ CPn+a be a variety. We first localize the problem; we give a criterion for X to be a complete intersection that is testable at any smooth point of X. We rephrase the criterion in the language of projec ..."
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Cited by 23 (14 self)
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Abstract. We characterize complete intersections in terms of local differential geometry. Let Xn ⊂ CPn+a be a variety. We first localize the problem; we give a criterion for X to be a complete intersection that is testable at any smooth point of X. We rephrase the criterion in the language of projective differential geometry and derive a sufficient condition for X to be a complete intersection that is computable at a general point x ∈ X. The sufficient condition has a geometric interpretation in terms of restrictions on the spaces of osculating hypersurfaces at x. When this sufficient condition holds, we are able to define systems of partial differential equations that generalize the classical Monge equation that characterizes conic curves in CP2. Using our sufficent condition, we show that if the ideal of X is generated by quadrics and a < n−(b+1)+3 3, where b =dimXsing, then X is a complete intersection. Local and global geometry
Projective planes, Severi varieties and spheres
, 2002
"... A classical result asserts that the complex projective plane modulo complex conjugation is the 4dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four project ..."
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Cited by 20 (1 self)
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A classical result asserts that the complex projective plane modulo complex conjugation is the 4dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.
Inductive characterizations of hyperquadrics
 Math. Ann
, 2008
"... Abstract. We give two characterizations of hyperquadrics: one as nondegenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as LQELmanifolds with large secant defects. 1. ..."
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Cited by 11 (1 self)
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Abstract. We give two characterizations of hyperquadrics: one as nondegenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as LQELmanifolds with large secant defects. 1.
The theta divisor of the bidegree (2,2) threefold
 in P 2 ×P 2 . preprint
, 1994
"... (0.1). In this paper we apply a new approach to the study of the theta divisor of a standard conic bundle. As an example we examine the Verra threefold T = T(2, 2) – the divisor of bidegree (2,2) in P 2 ×P 2. The threefold T deserves a special attention because of the recent observation of A.Verra ..."
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Cited by 2 (2 self)
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(0.1). In this paper we apply a new approach to the study of the theta divisor of a standard conic bundle. As an example we examine the Verra threefold T = T(2, 2) – the divisor of bidegree (2,2) in P 2 ×P 2. The threefold T deserves a special attention because of the recent observation of A.Verra (see [Ve]) that the existence of two conic bundle structures
Equations and syzygies of the first secant variety to a smooth curve
 Proc. Amer. Math. Soc
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