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24
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 19 (10 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
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.
Instanton sheaves on complex projective spaces
, 2008
"... We study a class of torsionfree sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable, while semistable torsionfree sheaves satisfying certain cohomolo ..."
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Cited by 4 (3 self)
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We study a class of torsionfree sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable, while semistable torsionfree sheaves satisfying certain cohomological conditions are instanton. We
Complex Algebraic Varieties and their Cohomology
"... This book evolved from my notes for assorted second semester courses in algebraic geometry given at Purdue. These are meant to give a rapid introduction to complex algebraic geometry for graduate students or possibly mathematicians in other fields. I assumed an understanding of basic algebraic geome ..."
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Cited by 3 (0 self)
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This book evolved from my notes for assorted second semester courses in algebraic geometry given at Purdue. These are meant to give a rapid introduction to complex algebraic geometry for graduate students or possibly mathematicians in other fields. I assumed an understanding of basic algebraic geometry (around the
Heights of ideals of minors
 In preparation
"... Abstract. We prove new height inequalities for determinantal ideals in a regular local ring, or more generally in a local ring of given embedding codimension. Our theorems extend and sharpen results of Faltings and Bruns. Introduction. Let ϕ be a map of vector bundles on a variety X. A wellknown the ..."
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Cited by 2 (1 self)
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Abstract. We prove new height inequalities for determinantal ideals in a regular local ring, or more generally in a local ring of given embedding codimension. Our theorems extend and sharpen results of Faltings and Bruns. Introduction. Let ϕ be a map of vector bundles on a variety X. A wellknown theorem of Eagon and Northcott [EN] gives an upper bound for the codimension of the locus where ϕ has rank ≤ s for any integer s. Bruns [B] improved this result by taking into account the generic rank r of ϕ. We shall see below that unlike the EagonNorthcott estimate, in most cases
Projections from Subvarieties
 De Gruyter
, 1998
"... This article started as an attempt to understand the structure of this mapping when / ..."
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Cited by 1 (1 self)
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This article started as an attempt to understand the structure of this mapping when /
On quadratic and higher normality of small codimension projective varieties (arXiv:math.AG/0002140
"... Abstract. Ran proved that smooth codimension 2 varieties in Pm+2 are jnormal if (j + 1)(3j − 1) ≤ m − 1, in this paper we extend this result to small codimension projective varieties. Let X be a r codimension subvariety of Pm+r, we prove that if the set Σ (j+1) of (j+1)secants to X through a gene ..."
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Abstract. Ran proved that smooth codimension 2 varieties in Pm+2 are jnormal if (j + 1)(3j − 1) ≤ m − 1, in this paper we extend this result to small codimension projective varieties. Let X be a r codimension subvariety of Pm+r, we prove that if the set Σ (j+1) of (j+1)secants to X through a generic external point is not empty, 2(r+1)j ≤ m−r and (j +1)((r+1)j −1) ≤ m−1 then X is jnormal. If X is given by the zero locus of a section of a rank r vector bundle E on P m+r, we prove that deg Σj+1 = 1 (j+1)! ∏ j i=0 cr(E(−i)). Moreover we get a new simple proof of Zak’s theorem on linear normality if
ON DOUBLE VERONESE EMBEDDINGS IN THE GRASSMANNIAN G(1, N)
, 2004
"... Abstract. We classify all the embeddings of P n in a Grassmannian of lines G(1, N) such that the composition with Plücker is given by a linear system of quadrics of P n. ..."
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Abstract. We classify all the embeddings of P n in a Grassmannian of lines G(1, N) such that the composition with Plücker is given by a linear system of quadrics of P n.
ON THE PICARD GROUP OF LOWCODIMENSION SUBVARIETIES
, 2005
"... Abstract. We introduce a method to determine if ndimensional smooth subvarieties of an ambient space of dimension at most 2n − 2 inherit the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, ..."
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Abstract. We introduce a method to determine if ndimensional smooth subvarieties of an ambient space of dimension at most 2n − 2 inherit the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, we give an affirmative answer (up to some mild natural numerical conditions) when the ambient space is a Grassmannian of lines (thus improving results of Barth, Van de Ven and Sommese) or a product of two projective spaces of the same dimension. A (complex) smooth subvariety of small codimension is expected to be very special. The most characteristic example of this principle is the case of subvarieties of the projective space, in which a famous conjecture of Hartshorne (see [7]) states that a smooth ndimensional subvariety X ⊂ P N must be a complete intersection if 2N < 3n. The main evidence for this conjecture is a series of results of Barth and Larsen showing that the topology of X behaves like the one of a complete intersection. Specifically, it was first proved in [3] that H i (X, Q) ∼ = H i (P N, Q) if