Results 1  10
of
16
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, ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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
On the branch curve of a general projection of a surface to a plane
, 2008
"... 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. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
Tight immersions and local differential geometry
, 1997
"... Historically, the study of tight immersions of manifolds had its origins in the study of immersions with minimal total absolute curvature. The most significant result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which completely characterizes minima ..."
Abstract
 Add to MetaCart
Historically, the study of tight immersions of manifolds had its origins in the study of immersions with minimal total absolute curvature. The most significant result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which completely characterizes minimal total absolute curvature immersions (and tight immersions) of spheres into a Euclidean space. An essential ingredient in this characterization was a reformulation of the problem in terms of the Morse theory of linear height functions and the topological characteristics of the manifold being immersed. In this sense, the theorem of Chern and Lashof characterizes tight immersions of the topologically simplest compact manifold. It is very natural to try and characterize tight immersions of other manifolds with topological restrictions. The most natural candidates are highly connected manifolds; i.e., 2kdimensional manifolds that are (k − 1)connected, but not kconnected. In the years since the theorem of Chern and Lashof was proved, tight immersions of these manifolds have been studied extensively by Kuiper and others. The purpose of the present paper is to finish a program begun by Kuiper and complete the proof of the following theorem:
The universal rank(n − 1) bundle on G(1,n) restricted to subvarieties
, 1997
"... En record de Ferran Serrano, extraordinari com a persona i com a matemàtic We classify those smooth (n − 1)folds in G(1, P n) for which the restriction of the rank(n − 1) universal bundle has more than n + 1 independent sections. As an aplication, we classify also those (n − 1)folds for which tha ..."
Abstract
 Add to MetaCart
En record de Ferran Serrano, extraordinari com a persona i com a matemàtic We classify those smooth (n − 1)folds in G(1, P n) for which the restriction of the rank(n − 1) universal bundle has more than n + 1 independent sections. As an aplication, we classify also those (n − 1)folds for which that bundle splits. It is a classical problem to study which projective varieties of small codimension are not linearly normal (i.e. isomorphically projected from higher projective spaces). The first observation is that any ndimensional variety can be projected to P 2n+1, but is expected to produce singular points when projected to P 2n. Hence, ndimensional varieties of codimension at most n are expected to be linearly normal. For n = 2, Severi proved (see his classical paper [6]) that the Veronese surface is the only smooth surface in P 5 that can be isomorphically projected to P 4. In general, projectability (or linear normality) is characterized by the dimension of the secant varieties. A recent thorough study of secant varieties can be found in [7], where a lot of projectability
On the dimension of secant varieties
, 2008
"... In this paper we generalize Zak’s theorems on tangencies and on linear normality as well as Zak’s definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, ..."
Abstract
 Add to MetaCart
In this paper we generalize Zak’s theorems on tangencies and on linear normality as well as Zak’s definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.