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Induction for secant varieties of Segre varieties
"... Abstract. 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 reduc ..."
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Cited by 55 (13 self)
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Abstract. 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 psecant varieties to Segre varieties for p ≤ 6. We generalize a theorem of CatalisanoGeramitaGimigliano on nondefectivity of tensor powers of Pn. 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 P1 × P1 × Pn × Pn and P2 × P3 × P3. In the final section we propose a series of conjectures about defective Segre varieties. 1.
On the Alexander–Hirschowitz theorem
 J. Pure Appl. Algebra
, 2008
"... The AlexanderHirschowitz theorem says that a general collection of k double points in Pn imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, con ..."
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Cited by 39 (8 self)
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The AlexanderHirschowitz theorem says that a general collection of k double points in Pn imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a selfcontained proof based mainly on previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d = 3, where our proof is shorter. We end with an account of the history of the work on this problem.
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 35 (5 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.
DECOMPOSITION OF HOMOGENEOUS POLYNOMIALS WITH LOW RANK
 MATHEMATISCHE ZEITSCHRIFT
, 2011
"... Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the duple Veronese embedding of Pm into P (m+d d)−1 but that its minimal decomposition as a sum of dth powers of ..."
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Cited by 19 (13 self)
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Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the duple Veronese embedding of Pm into P (m+d d)−1 but that its minimal decomposition as a sum of dth powers of linear forms M1,..., Mr is F = M d 1 + · · ·+M d r with r> s. We show that if s+r ≤ 2d+1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.
Varieties with minimal secant degree and linear systems of maximal dimension on surfaces
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Classifying Hilbert functions of fat point subschemes in P²
, 2008
"... The paper [GMS] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1 + · · · + 2pr, for all possible choices of r distinct points in P 2. We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic ..."
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Cited by 17 (8 self)
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The paper [GMS] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1 + · · · + 2pr, for all possible choices of r distinct points in P 2. We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic in case either r ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [H2] and [H3] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say p1,..., pr and p ′ 1,..., p ′ r have the same configuration type if for all choices of nonnegative integers mi, Z = m1p1 + · · · + mrpr and Z ′ = m1p ′ 1 + · · · + mrp ′ r have the same Hilbert function.) Assuming either that 7 ≤ r ≤ 8 (see [GuH] for the cases r ≤ 6) or that the points pi lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients mi determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of Z = m1p1 + · · · + mrpr. We demonstrate our results by explicitly listing all Hilbert functions for schemes of r ≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.
Topics on Interpolation Problems in Algebraic Geometry
, 2004
"... These are notes of the lectures given by the authors during the school/workshop “Polynomial Interpolation and Projective Embeddings”. We mainly focus our attention on the planar case and on the Segre and HarbourneHirschowitz Conjectures. We discuss the state of the art introducing several results ..."
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Cited by 11 (3 self)
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These are notes of the lectures given by the authors during the school/workshop “Polynomial Interpolation and Projective Embeddings”. We mainly focus our attention on the planar case and on the Segre and HarbourneHirschowitz Conjectures. We discuss the state of the art introducing several results and different techniques.
Orecchia: Bivariate Hermite interpolation and linear systems of plane curves with base fat points
 Computer mathematics, Lecture Notes Ser. Comput., 10, World Sci. Publishing, River Edge, NJ, (2003) 87–102. of the Veronese and Applications 797
"... It is still an open question to determine in general the dimension of the vector space of bivariate polynomials of degree at most d which have all partial derivatives up through order mi − 1 vanish at each point pi (i = 1,..., n), for some fixed integer mi called multiplicity at pi. When the multip ..."
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It is still an open question to determine in general the dimension of the vector space of bivariate polynomials of degree at most d which have all partial derivatives up through order mi − 1 vanish at each point pi (i = 1,..., n), for some fixed integer mi called multiplicity at pi. When the multiplicities are all equal, to m say, this problem has been attacked by a number of authors (Lorentz and Lorentz, Ciliberto and Miranda, Hirschowitz) and there are a number of good conjectures (Hirschowitz, Ciliberto and Miranda) on the dimension of these interpolating spaces. The determination of the dimension has been already solved for m ≤ 12 and all d and n by a degeneration technique and some ad hoc geometric arguments. Here this technique is applied up through m = 20; since it fails in some cases, we resort (in these exceptional cases) to the bivariete Hermite interpolation with the support of a simple idea suggested by Gröbner bases computation. In summary we are able to prove that the dimension of the vector space is the expected one for 13 ≤ m ≤ 20. 1.
COMPLETE INTERSECTIONS ON GENERAL HYPERSURFACES
, 801
"... Abstract. We ask when certain complete intersections of codimension r can lie on a generic hypersurface in P n. We give a complete answer to this question when 2r ≤ n + 2 in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection. 1. ..."
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Cited by 11 (8 self)
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Abstract. We ask when certain complete intersections of codimension r can lie on a generic hypersurface in P n. We give a complete answer to this question when 2r ≤ n + 2 in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection. 1.