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14
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.
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 12 (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.
Flamini On the branch curve of a general projection of a surface to a plane, arXiv:0811.0467v1
, 2008
"... Abstract. In this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible, with only nodes and cusps. 1. ..."
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Cited by 2 (0 self)
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Abstract. In this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible, with only nodes and cusps. 1.
VARIETIES WITH QUADRATIC ENTRY LOCUS, I
, 2007
"... We shall introduce and study quadratic entry locus varieties, a class of projective algebraic varieties whose extrinsic and intrinsic geometry is very rich. Let us recall that, for an irreducible nondegenerate variety X ⊂ P N of dimension n ≥ 1, the secant defect of X, denoted by δ(X), is the diffe ..."
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Cited by 2 (1 self)
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We shall introduce and study quadratic entry locus varieties, a class of projective algebraic varieties whose extrinsic and intrinsic geometry is very rich. Let us recall that, for an irreducible nondegenerate variety X ⊂ P N of dimension n ≥ 1, the secant defect of X, denoted by δ(X), is the difference between the expected dimension and the effective dimension of the
Fontanari: Birational geometry of defective varieties
, 2003
"... Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectivity, and rationality. AMS Subject Classif ..."
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Cited by 1 (1 self)
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Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectivity, and rationality. AMS Subject Classification: 14N05.
SOME EXTREMAL CONTRACTIONS BETWEEN SMOOTH VARIETIES ARISING FROM PROJECTIVE GEOMETRY
, 2004
"... Let ψ: X → Y be a proper morphism with connected fibers from a smooth projective variety X onto a normal variety Y, i.e. a contraction. If −KX is ψample, then ψ is said to be an extremal contraction and if moreover Pic(X)/ψ ∗ (Pic(Y)) ≃ Z then ψ is said to be an elementary extremal contraction or a ..."
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Cited by 1 (0 self)
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Let ψ: X → Y be a proper morphism with connected fibers from a smooth projective variety X onto a normal variety Y, i.e. a contraction. If −KX is ψample, then ψ is said to be an extremal contraction and if moreover Pic(X)/ψ ∗ (Pic(Y)) ≃ Z then ψ is said to be an elementary extremal contraction or a FanoMori contraction.
Rational And NonRational Algebraic Varieties: Lectures Of János Kollár
, 1997
"... Introduction Rational varieties are among the simplest possible algebraic varieties. Their study is as old as algebraic geometry itself, yet it remains a remarkably difficult area of research today. These notes offer an introduction to the study of rational varieties. We begin with the beautiful cl ..."
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Introduction Rational varieties are among the simplest possible algebraic varieties. Their study is as old as algebraic geometry itself, yet it remains a remarkably difficult area of research today. These notes offer an introduction to the study of rational varieties. We begin with the beautiful classical geometric approach to finding examples of rational varieties, and end with some subtle algebraic arguments that have recently established nonrationality of varieties that otherwise share many of their traits. In lecture one, rationality and unirationality are defined, and illustrated with a series of examples. We also introduce some easily computable invariants, the plurigenera, that vanish for all rational varieties. Using the plurigenera, one develops a sense how rare rational varieties are. For example, we will immediately see that no smooth projective hypersurface whose degree exceeds its embedding dimension can be rational: no plane cubic curve is rational, no space qu
VARIETIES WITH QUADRATIC ENTRY LOCUS, II
, 2008
"... Abstract. We continue the study, begun in [Ru07], of secant defective manifolds having “simple entry loci”. We prove that such manifolds are rational and describe them in terms of tangential projections. Using also [IR07], their classification is reduced to the case of Fano manifolds of high index, ..."
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Abstract. We continue the study, begun in [Ru07], of secant defective manifolds having “simple entry loci”. We prove that such manifolds are rational and describe them in terms of tangential projections. Using also [IR07], their classification is reduced to the case of Fano manifolds of high index, whose Picard group is generated by the hyperplane section class. Conjecturally, the former should be linear sections of rational homogeneous manifolds. We also provide evidence that the classification of linearly normal dual defective manifolds with Picard group generated by the hyperplane section should follow along the same lines.
Tangential projections and secant defective varieties
, 2008
"... Abstract: Going one step further in Zak’s classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of P n given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and nondegenerat ..."
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Abstract: Going one step further in Zak’s classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of P n given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and nondegenerate projective subvarieties of PN that can be projected isomorphically into P2n when N ≥ − 2.