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23
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
, 2001
"... In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposi ..."
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Cited by 56 (26 self)
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In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As byproducts, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.
Syzygies, multigraded regularity and toric varieties
, 2006
"... Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1,...,Bℓ on X and m1,...,mℓ ∈ N, consider the line bundle L: = B m1 1 ⊗···⊗Bmℓ ℓ. We give conditions on the mi which guarantee that the ide ..."
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Cited by 15 (6 self)
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Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1,...,Bℓ on X and m1,...,mℓ ∈ N, consider the line bundle L: = B m1 1 ⊗···⊗Bmℓ ℓ. We give conditions on the mi which guarantee that the ideal of X in P(H0 (X, L) ∗ ) is generated by quadrics and that the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
Vector bundles on curves and theta functions
, 2005
"... These notes survey the relation between the moduli spaces of vector bundles on a curve C and the spaces of (classical) theta functions on the Jacobian J of C. The connection appears when one tries to describe the moduli space Mr of rank r ..."
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Cited by 13 (0 self)
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These notes survey the relation between the moduli spaces of vector bundles on a curve C and the spaces of (classical) theta functions on the Jacobian J of C. The connection appears when one tries to describe the moduli space Mr of rank r
On Cohomology of the Square of an Ideal Sheaf
 Address for Offprints: Department of Mathematics, Oklahoma State University, Stillwater
, 1997
"... Abstract. For a smooth subvariety X ⊂ PN, consider (analogously to projective (k)) = 0, k ≥ 3. This condition is normality) the vanishing condition H1 (PN, I2 X shown to be satisfied for all sufficiently large embeddings of a given X, and for a Veronese embedding of Pn. For C ⊂ Pg−1, the canonical ..."
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Cited by 11 (0 self)
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Abstract. For a smooth subvariety X ⊂ PN, consider (analogously to projective (k)) = 0, k ≥ 3. This condition is normality) the vanishing condition H1 (PN, I2 X shown to be satisfied for all sufficiently large embeddings of a given X, and for a Veronese embedding of Pn. For C ⊂ Pg−1, the canonical embedding of a nonhyperelliptic curve, this condition guarantees the vanishing of some obstruction groups to deformations of the cone. Recall that the tangents to deformations are dual to the cokernel of the GaussianWahl map. Theorem. Suppose the GaussianWahl map of C is not surjective and the vanishing condition is fulfilled. Then C is extendable: it is a hyperplane section of a surface in P g not the cone over C. Such a surface is a K3 if smooth, but it could have serious singularities. Theorem. For a general curve of genus ≥ 3, this vanishing holds. Conjecture. If the Clifford index is ≥ 3, this vanishing holds.
Seshadri constants, gonality of space curves, and restriction of stable bundles
 J. Differential Geom
, 1994
"... There exist many situations in algebraic geometry where the extrinsic geometry of a variety is reflected in clear restrictions in the way that it can map to projective spaces. For example, it is wellknown that the gonality of a smooth plane curve C of degree d is d − 1, and that ..."
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Cited by 8 (0 self)
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There exist many situations in algebraic geometry where the extrinsic geometry of a variety is reflected in clear restrictions in the way that it can map to projective spaces. For example, it is wellknown that the gonality of a smooth plane curve C of degree d is d − 1, and that
Example to “Determinantal equations for secant varieties and the EisenbudKohStillman conjecture
, 2012
"... Abstract. We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese reembeddings of arbitrary varieties. Eisenbud’s question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that settheoretic equations of small secant varietie ..."
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Cited by 6 (0 self)
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Abstract. We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese reembeddings of arbitrary varieties. Eisenbud’s question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that settheoretic equations of small secant varieties to a high degree Veronese reembedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counterexamples to the EKS conjecture for singular curves. The techniques we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbud’s question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provide an overview of conjectures coming from signal processing and complexity theory in this context. 1.
The Ideal Generation Problem for Fat Points
 J. Pure Appl. Alg
, 2000
"... Abstract: This paper is concerned with the problem of determining up to graded isomorphism the modules in a minimal free resolution of a fat point subscheme Z = m1p1 + · · · + mrpr ⊂ P 2 for general points p1,..., pr. We always work over an arbitrary algebraically closed field k. This paper is con ..."
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Cited by 5 (4 self)
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Abstract: This paper is concerned with the problem of determining up to graded isomorphism the modules in a minimal free resolution of a fat point subscheme Z = m1p1 + · · · + mrpr ⊂ P 2 for general points p1,..., pr. We always work over an arbitrary algebraically closed field k. This paper is concerned with determining the number νt(I(Z)) of elements in each degree t of any minimal set of homogeneous generators in the ideal I(Z) ⊂ k[P 2] defining a fat point subscheme Z = m1p1+ · · ·+mrpr ⊂ P 2, where p1,...,pr ∈ P 2 are general. Given the Hilbert function of I(Z), this is equivalent up to graded isomorphism to determining the modules
Embeddings of homogeneous spaces in prime characteristics
 Amer. J. Math
, 1996
"... Let X be a projective algebraic variety over an algebraically closed field k admitting a homogeneous action of a semisimple linear algebraic group G. Then X can be canonically identified with the homogeneous space G/Gx, where x is a closed point in X and Gx the stabilizer group scheme of x. A group ..."
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Cited by 2 (1 self)
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Let X be a projective algebraic variety over an algebraically closed field k admitting a homogeneous action of a semisimple linear algebraic group G. Then X can be canonically identified with the homogeneous space G/Gx, where x is a closed point in X and Gx the stabilizer group scheme of x. A group scheme over a field of characteristic 0 is reduced so in this case, X is isomorphic to a generalized flag variety G/P, where P is a parabolic subgroup. In [2][7][6][5] the geometry of X in prime characteristic has been studied and it has been shown that a lot of strange phenomena occur when Gx is nonreduced. The simplest example of a projective homogeneous Gspace (for G = SL3(k), char k = p> 0) not isomorphic to a generalized flag variety is the divisor x0 y p 0 + x1 y p 1 + x2 y p 2 = 0 in P 2 × P 2. Since projective homogeneous spaces with nonreduced stabilizers are quite algebraic by construction, we give in §2 of this paper a simple geometric approach for their construction, involving only scheme theoretic images under partial Frobenius morphisms. We choose to do this focusing on the “unseparated incidence variety”.
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 /