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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.
On codimension two subvarieties of P 6.
, 1999
"... One of the most attractive problems in algebraic geometry is Hartshorne’s conjecture ([9]): ”let X ⊂ P n (C) be a smooth subvariety, if dim(X)> ..."
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One of the most attractive problems in algebraic geometry is Hartshorne’s conjecture ([9]): ”let X ⊂ P n (C) be a smooth subvariety, if dim(X)>
MANIFOLDS COVERED BY LINES, DEFECTIVE MANIFOLDS AND A RESTRICTED HARTSHORNE CONJECTURE
, 909
"... ABSTRACT. Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for ma ..."
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ABSTRACT. Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds. Using the geometry of the variety of lines through a general point, we characterize scrolls among dual defective manifolds. This leads to an optimal bound for the dual defect, which improves results due to Ein. We discuss our conjecture that every dual defective manifold with cyclic Picard group should also be secant defective, of a very special type, namely a local quadratic entry locus variety.
Codimension two nonsingular subvarieties of quadrics: scrolls and classification in degree d ≤ 10
, 1996
"... Let X be a codimension two nonsingular subvariety of a nonsingular quadric Q n of dimension n ≥ 5. We classify such subvarieties when they are scrolls. We also classify them when the degree d ≤ 10. Both results were known when n = 4. ..."
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Let X be a codimension two nonsingular subvariety of a nonsingular quadric Q n of dimension n ≥ 5. We classify such subvarieties when they are scrolls. We also classify them when the degree d ≤ 10. Both results were known when n = 4.
Some adjunctiontheoretic properties of codimension two nonsingular subvarieties of quadrics
, 1996
"... We make precise the structure of the first two reduction morphisms associated with codimension two nonsingular subvarieties of quadrics Q n, n ≥ 5. We give a coarse classification of the same class of subvarieties when they are assumed to be not of loggeneraltype. ..."
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We make precise the structure of the first two reduction morphisms associated with codimension two nonsingular subvarieties of quadrics Q n, n ≥ 5. We give a coarse classification of the same class of subvarieties when they are assumed to be not of loggeneraltype.
MANIFOLDS COVERED BY LINES AND THE HARTSHORNE CONJECTURE FOR QUADRATIC MANIFOLDS
"... ABSTRACT. Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for ma ..."
Abstract
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ABSTRACT. Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds.