Linear and Steiner Bundles on Projective Varieties

doi:10.1080/00927871003757584

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Linear and Steiner Bundles on Projective Varieties

Journal Article•DOI•

Linear and Steiner Bundles on Projective Varieties

Marcos Jardim¹, Renato Vidal Martins²•Institutions (2)

State University of Campinas¹, Universidade Federal de Minas Gerais²

14 Jun 2010-Communications in Algebra (Taylor & Francis Group)-Vol. 38, Iss: 6, pp 2249-2270

TL;DR: In this article, the authors generalize the theory of Horrocks monads to ACM varieties, and use the generalization to establish a cohomological characterization of linear and Steiner bundles on projective space and on quadric hypersurfaces.

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Abstract: We generalize the theory of Horrocks monads to ACM varieties, and use the generalization to establish a cohomological characterization of linear and Steiner bundles on projective space and on quadric hypersurfaces. We also characterize Steiner bundles on the Grassmannian G(1, 4) of lines in ℙ4. Finally, we study linear resolutions of bundles on ACM varieties, and characterize linear homological dimension on quadric hypersurfaces.

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Journal Article•DOI•

ADHM construction of perverse instanton sheaves

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Abdelmoubine Amar Henni¹, Marcos Jardim², Renato Vidal Martins³•Institutions (3)

Universidade Federal de Santa Catarina¹, State University of Campinas², Universidade Federal de Minas Gerais³

01 May 2015-Glasgow Mathematical Journal

TL;DR: In this paper, a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line is presented, which further generalises the one on projective spaces.

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Abstract: We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalises the one on projective spaces. This is done by generalising the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterisation of these sheaves in the particular case of projective spaces.

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24 citations

Journal Article•DOI•

On the singular scheme of split foliations

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Marcos Jardim, Maurício Corrêa, Renato Vidal Martins

01 Jan 2015-Indiana University Mathematics Journal

TL;DR: In this article, it was shown that the tangent sheaf of a codimension one locally free distribution splits as a sum of line bundles if and only if its singular scheme is arithmetically Cohen-Macaulay.

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Abstract: We prove that the tangent sheaf of a codimension one locally free distribution splits as a sum of line bundles if and only if its singular scheme is arithmetically Cohen-Macaulay. In addition, we show that a foliation by curves is given by an intersection of generically transversal holomorphic distributions of codimension one if and only if its singular scheme is arithmetically Buchsbaum. Finally, we establish that these foliations are determined by their singular schemes, and deduce that the Hilbert scheme of certain arithmetically Buchsbaum schemes of codimension $2$ is birational to a Grassmannian.

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21 citations

Journal Article•DOI•

The ADHM variety and perverse coherent sheaves

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Marcos Jardim¹, Renato Vidal Martins²•Institutions (2)

State University of Campinas¹, Universidade Federal de Minas Gerais²

01 Nov 2011-Journal of Geometry and Physics

TL;DR: In this article, the full set of solutions to the ADHM equation is studied as an affine algebraic set, and a filtration of these solutions into subvarieties according to the dimension of the stabilizing subspace is shown.

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8 citations

Posted Content•

New moduli components of rank 2 bundles on projective space

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Charles Almeida, Marcos Jardim, Alexander S. Tikhomirov, Sergey Tikhomirov

21 Feb 2017-arXiv: Algebraic Geometry

TL;DR: In this paper, a new family of monads whose cohomology is a stable rank two vector bundle is presented, and the irreducibility and smoothness of these monads are studied.

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Abstract: We present a new family of monads whose cohomology is a stable rank two vector bundle on $\mathbb{P}^3$. We also study the irreducibility and smoothness together with a geometrical description of some of these families. These facts are used to construct a new infinite series of rational moduli components of stable rank two vector bundles with trivial determinant and growing second Chern class. We also prove that the moduli space of stable rank two vector bundles with trivial determinant and second Chern class equal to 5 has exactly three irreducible rational components.

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8 citations

Journal Article•DOI•

Differential geometry of Hilbert schemes of curves in a projective space

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Roger Bielawski¹, Carolin Peternell¹•Institutions (1)

Leibniz University of Hanover¹

01 Jan 2019-Complex Manifolds

TL;DR: In this article, the natural geometry of Hilbert schemes of curves is described in terms of a Hilbert scheme of curves in √ √ n σ 2 σ 3, √ σ 4 σ σ n 2, where σ is the number of vertices.

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Abstract: We describe the natural geometry of Hilbert schemes of curves in ${\mathbb P}^3$ and, in some cases, in ${\mathbb P}^n$ , $n\geq 4$.

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6 citations

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Book•

Vector Bundles on Complex Projective Spaces

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Christian Okonek, Michael Schneider, Heinz Spindler

01 Jan 1980

1,339 citations

"Linear and Steiner Bundles on Proje..." refers background or methods in this paper

...ed the study of some special kind of monads which were named Horrocks later on (cf. [1] and [5] for example). Monads on projective spaces have been much studied in the past 25 years (see for instance [15, 17, 21, 22] and the references therein). More recently, many authors have also been interested on monads over more general varieties, see [8, 11, 18]. We start the present article in Section 2 studying general m...
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... tool in the cohomological characterization of linear and Steiner bundles. We devote Section 4 to the study of linear bundles. Linear bundles on P2 and P3 have been studied since the late 1970’s (cf. [22]), and the mathematical instanton bundles on P2n+1 introduced by Okonek and Spindler in [23] are examples of linear bundles. More general linear monads were ﬁrst considered in [18]. A cohomological ch...
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Journal Article•DOI•

Cohen-Macaulay modules on hypersurface singularities I

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Horst Knörrer¹•Institutions (1)

University of Bonn¹

01 Feb 1987-Inventiones Mathematicae

456 citations

Journal Article•DOI•

Vector Bundles on the Punctured Spectrum of a Local Ring

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G. Horrocks¹•Institutions (1)

University of Liverpool¹

01 Oct 1964-Proceedings of The London Mathematical Society

323 citations

"Linear and Steiner Bundles on Proje..." refers background in this paper

...Monads were introduced in the 60s by Horrocks [13]....
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Journal Article•DOI•

Cohen-Macaulay modules on hypersurface singularities II

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Ragnar-Olaf Buchweitz, Gert-Martin Greuel, Frank-Olaf Schreyer

01 Feb 1987-Inventiones Mathematicae

TL;DR: In this paper, the authors present a generalization of matrix factorizations to matrix factorization on A~ and D. The results show that the matrix factorisation on D can be expressed as a matrix decomposition of a periodic complex.

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Abstract: 0. Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 l. Periodic complexes and matrix factorizations . . . . . . . . . . . . . . . . . . . . 169 2. Construction of matrix factorizations and MCM's . . . . . . . . . . . . . . . . . . 173 3. Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4. MCM's on A~ and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

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304 citations

"Linear and Steiner Bundles on Proje..." refers background in this paper

... hand, hyperplanes and quadrics are the only hypersurfaces in projective space for which there are only a ﬁnite number of indecomposable locally-free ACM sheaves, up to twisting by a line bundle (cf. [9]). Some speciﬁc varieties have been looked at in the literature; in [3] the authors classify all locally-free ACM sheaves on the grassmannian of lines in P4; certain Fano 3-folds were considered in [4...
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Journal Article•DOI•

Spinor bundles on quadrics

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Giorgio Ottaviani

01 Jan 1988-Transactions of the American Mathematical Society

TL;DR: In this article, the authors define stable vector bundles on the complex quadric hypersurface Qn of dimension n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3).

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Abstract: We define some stable vector bundles on the complex quadric hypersurface Qn of dimension n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. When n = 2fc — 1 there is one spinor bundle of rank 2k~1. When n = 2k there are two spinor bundles of rank 2k~1. Their behavior is slightly different according as n = 0 (mod 4) or n = 2 (mod 4). As an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qe- Introduction. Let Qn be the smooth quadric hypersurface of the complex pro- jective space Pn+1. In this paper we define in a geometrical way some vector bundles on the quadric Qn as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. On Q4 this definition is equivalent to the usual one. Spinor bundles are homogeneous and stable (according to the definition of Mum- ford-Takemoto). We study their first properties using the geometrical description given and some standard techniques available in (OSS). We also use a theorem of Ramanan (see (Um)) about the stability of homoge- neous bundles induced by irreducible representations. When n is odd there is only one spinor bundle, while when n is even there are two nonisomorphic spinor bun- dles. When n is even the behavior of spinor bundles is slightly different according as n = 0 (mod4) or n = 2 (mod4). In (Ot2) we have given a cohomological splitting criterion for vector bundles on quadrics involving spinor bundles. Qn ~ Spin(n + 2)/P(cty) (St) is a homogeneous manifold, and the semisimple part of the Lie algebra of -P(ai) is o(n). At the level of Lie algebras, spinor bundles are defined from the spin and half-spin representations of o(n). The paper is divided into three sections. In §1 we give some preliminary results and we define the spinor bundles. In §2 we study the first properties of spinor bundles. In §3, as an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qq.

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173 citations

"Linear and Steiner Bundles on Proje..." refers background or methods in this paper

...We will use some basic facts regarding spinor bundles on quadrics, see [24, 25 ]....
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...Proof. From Proposition 3.8 we have that if E is the cohomology of the stated monad then (i)-(v) hold because OQn is an ACM sheaf with parameters s = −1 and t = 1 −n and the spinor bundles � are ACM sheaves with parameters s = 0 and t = 1 −n (cf. [ 25, Theorem 2.8 ])....
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