Results 1  10
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75
Multigraded Hilbert schemes
 J. Algebraic Geom
"... We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, includ ..."
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Cited by 63 (3 self)
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We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer’s conjecture on equations defining Grothendieck’s classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes. 1.
Hyperplane arrangement cohomology and monomials in the exterior algebra
, 2003
"... We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of Aramova, ..."
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Cited by 29 (4 self)
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We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.
The resultant of an unmixed bivariate system
 J. of Symbolic Computation
"... This paper gives an explicit method for computing the resultant of any sparse unmixed bivariate system with given support. We construct square matrices whose determinant is exactly the resultant. The matrices constructed are of hybrid Sylvester and Bézout type. The results extend those in [14] by gi ..."
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Cited by 23 (1 self)
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This paper gives an explicit method for computing the resultant of any sparse unmixed bivariate system with given support. We construct square matrices whose determinant is exactly the resultant. The matrices constructed are of hybrid Sylvester and Bézout type. The results extend those in [14] by giving a complete combinatorial description of the matrix. Previous work by D’Andrea [5] gave pure Sylvester type matrices (in any dimension). In the bivariate case, D’Andrea and Emiris [7] constructed hybrid matrices with one Bézout row. These matrices are only guaranteed to have determinant some multiple of the resultant. The main contribution of this paper is the addition of new Bézout terms allowing us to achieve exact formulas. We make use of the exterior algebra techniques of Eisenbud, Fløystad, and Schreyer [10, 9]. 1.
Resonance, linear syzygies, Chen groups, and the BernsteinGelfandGelfand correspondence
 TRANS. AMER. MATH. SOC
, 2006
"... Abstract. If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H∗(X, k), viewed as a module over the exterior algebra E on A: θ_k(G) = ..."
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Cited by 21 (11 self)
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Abstract. If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H∗(X, k), viewed as a module over the exterior algebra E on A: θ_k(G) =dim_k Tor^E_{k−1} (A, k)_k, for k ≥ 2, where k is a field of characteristic 0. The Chen ranks conjecture asserts that, for k sufficiently large, θ_k(G) =(k − 1) ∑ r≥1 h_r \binom{r+k−1}{k}, where h_r is the number of rdimensional components of the projective resonance variety R_1 (A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R_1 (A) and a localization argument, we establish the inequality θ_k(G) ≥ (k − 1) ∑ \binom{r + k − 1}{k} h_r, for k ≫ 0, for arbitrary A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R_1 (A), such that θ_k(G) =P(k), for all k ≫ 0.
Derived category of squarefree modules and local cohomology with monomial ideal support
 J. MATH. SOC. JAPAN
, 2003
"... A squarefree module over a polynomial ring S = k[x1,..., xn] is a generalization of a StanleyReisner ring, and allows us to apply homological methods to the study of monomial ideals more systematically. ..."
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Cited by 17 (10 self)
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A squarefree module over a polynomial ring S = k[x1,..., xn] is a generalization of a StanleyReisner ring, and allows us to apply homological methods to the study of monomial ideals more systematically.
Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds
 Invent. Math
"... Given an irregular compact Kähler manifold X, one can form the derivative complex of X, which governs the deformation theory of the groups H i(X,α) as α varies over Pic0(X). Together with its variants, it plays a central role in a body of work involving generic vanishing theorems. The purpose of th ..."
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Cited by 16 (4 self)
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Given an irregular compact Kähler manifold X, one can form the derivative complex of X, which governs the deformation theory of the groups H i(X,α) as α varies over Pic0(X). Together with its variants, it plays a central role in a body of work involving generic vanishing theorems. The purpose of this paper is to present two new applications of this complex. First, we show that
Sheaf algorithms using the exterior algebra
, 2002
"... In this chapter we explain constructive methods for computing the cohomology of a sheaf on a projective variety. We also give a construction for the Beilinson monad, a tool for studying the sheaf from partial knowledge of its cohomology. Finally, we give some examples illustrating the use of the Bei ..."
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Cited by 15 (0 self)
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In this chapter we explain constructive methods for computing the cohomology of a sheaf on a projective variety. We also give a construction for the Beilinson monad, a tool for studying the sheaf from partial knowledge of its cohomology. Finally, we give some examples illustrating the use of the Beilinson monad.