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Induction for secant varieties of Segre varieties
 Trans. Amer. Math. Soc
"... This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calc ..."
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Cited by 25 (4 self)
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This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective tsecant varieties to Segre varieties for t ≤ 6. We generalize a theorem of CatalisanoGeramitaGimigliano on nondefectivity of tensor powers of P n. We determine the set of p for which unbalanced Segre varieties have defective psecant varieties. In addition, we completely describe the dimensions of the secant varieties to the deficient Segre varieties P 1 ×P 1 ×P n ×P n and P 2 × P 3 × P 3. In the final section we propose a series of conjectures about
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.
Monica: Computing symmetric rank for symmetric tensors
 J. Symbolic Comput
"... We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for 2 × · · · × 2 tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetri ..."
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Cited by 9 (4 self)
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We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for 2 × · · · × 2 tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties. Key words: Symmetric tensor, tensor rank, secant variety. 1.
ON SECANT VARIETIES OF COMPACT HERMITIAN SYMMETRIC SPACES
"... Abstract. We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three with one exception, the secant variety of the 21dimensional spinor v ..."
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Cited by 5 (1 self)
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Abstract. We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three with one exception, the secant variety of the 21dimensional spinor variety in P 63, whose ideal is generated in degree four. We also discuss the coordinate ring of secant varieties of compact Hermitian symmetric spaces. 1.
Brambilla, Secant varieties of SegreVeronese varieties P m × P n embedded by O(1
 arXiv (2008), no. 0809.4837. ↑2 DANIEL ERMAN AND MAURICIO VELASCO
"... Abstract. Let Xm,n be the SegreVeronese variety P m × P n embedded by the morphism given by O(1,2). In this paper, we provide two functions s(m, n) ≤ s(m, n) such that the s th secant variety of Xm,n has the expected dimension if s ≤ s(m,n) or s(m, n) ≤ s. We also present a conjecturally complete ..."
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Cited by 5 (0 self)
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Abstract. Let Xm,n be the SegreVeronese variety P m × P n embedded by the morphism given by O(1,2). In this paper, we provide two functions s(m, n) ≤ s(m, n) such that the s th secant variety of Xm,n has the expected dimension if s ≤ s(m,n) or s(m, n) ≤ s. We also present a conjecturally complete list of defective secant varieties of such SegreVeronese varieties. 1.
Secant dimensions of lowdimensional homogeneous varieties, Preprint arXiv:math.RT/:0707.1605
, 2007
"... Abstract. We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all SegreVeronese embeddings of P 1 × P 1, P 1 × P 1 × P 1, and P 2 × P 1, ..."
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Cited by 2 (0 self)
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Abstract. We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all SegreVeronese embeddings of P 1 × P 1, P 1 × P 1 × P 1, and P 2 × P 1, as well as for the variety F of incident pointline pairs in P 2. For P 2 × P 1 and F the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author’s tropical approach to secant dimensions. 1. Introduction and
RANKS OF TENSORS AND AND A GENERALIZATION OF SECANT VARIETIES
"... Abstract. We investigate differences between Xrank and Xborder rank, focusing on the cases of tensors and partially symmetric tensors. As an aid to our study, and as an object of interest in its own right, we define notions of Xrank and border rank for a linear subspace. Results include determini ..."
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Cited by 2 (0 self)
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Abstract. We investigate differences between Xrank and Xborder rank, focusing on the cases of tensors and partially symmetric tensors. As an aid to our study, and as an object of interest in its own right, we define notions of Xrank and border rank for a linear subspace. Results include determining and bounding the maximum Xrank of points in several cases of interest. 1.
LINEAR DETERMINANTAL EQUATIONS FOR ALL PROJECTIVE SCHEMES
, 2009
"... We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2×2 minors of a 1generic matrix of linear forms. Extending the work of EisenbudKohStillman for integral curves, we also provide effective ..."
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We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2×2 minors of a 1generic matrix of linear forms. Extending the work of EisenbudKohStillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth nfolds.
GENERATION AND SYZYGIES OF THE FIRST SECANT VARIETY
, 809
"... Abstract. Under certain effective positivity conditions, we show that the secant variety to a smooth variety satisfies N3,p. For smooth curves, we provide the best possible effective bound on the degree d of the embedding, d ≥ 2g + 3 + p. 1. ..."
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Abstract. Under certain effective positivity conditions, we show that the secant variety to a smooth variety satisfies N3,p. For smooth curves, we provide the best possible effective bound on the degree d of the embedding, d ≥ 2g + 3 + p. 1.
Mourrain, A comparison of different notions of ranks of symmetric tensors, Preprint: http://hal.inria.fr/hal00746967
"... Abstract. We introduce various notions of rank for a high order symmetric tensor,namely: rank,borderrank,catalecticantrank,generalizedrank,scheme length, borderschemelength, extensionrankandsmoothablerank. Weanalyze the stratification induced by these ranks. The mutual relations between these strati ..."
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Abstract. We introduce various notions of rank for a high order symmetric tensor,namely: rank,borderrank,catalecticantrank,generalizedrank,scheme length, borderschemelength, extensionrankandsmoothablerank. Weanalyze the stratification induced by these ranks. The mutual relations between these stratifications, allowus to describethe hierarchyamongall the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide. hal00746967, version 2 26 Nov 2012