Globally Generated Vector Bundles on P^n with c_1=4

Anghel, Cristian; Manolache, Nicolae

doi:10.48550/arxiv.1305.3464

Cited by 3 publications

(30 citation statements)

References 35 publications

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“…and so E admits the following resolution (1). Similarly when deg(Z) = 2, we obtain the resolution (2).…”

Section: Rank Two Globally Generated Vector Bundle Withmentioning

confidence: 72%

“…E fits into the sequence (4) with c 1 = (1, 2). Since I Z (1, 2) is globally generated, Z is contained in a complete intersection of two hypersurfaces of bidegree (1,2) in Q. In particular, deg(Z) is at most 4.…”

Section: Rank Two Globally Generated Vector Bundle Withmentioning

confidence: 99%

“…Globally generated vector bundles on projective varieties play an important role in projective algebraic geometry and their classification has been done quite recently over projective spaces by many people [2][3] [8][9] [15][17] [18]. Especially in [9], Ellia determines the Chern classes of globally generated vector bundles of rank 2 on P 2 , using several results on Lüroth semigroup of smooth plane curves [7] [10].…”

Section: Introductionmentioning

confidence: 99%

“…(1, 1, 2; r = 2, 3), (1, 2, 2; 2), (1, 2, 3; r = 2, 3), (1,2,4; r = 2, 3, 4, 5), (2, 2, 3; 2), (2,2,4; r = 2, 3), (2, 2, 5; r = 2, 3), (2,2,6; r = 2, 3, 4, 5), (2, 2, 8; r = 2, 3, 4, 5, 6, 7, 8) In the second section we fix the notations and we explain the preliminaries. In the third section we show that any globally generated vector bundle with c 1 = (0, 0) or c 1 = (a, 0) and either a ≤ 2 or rank two splits.…”

Section: Introductionmentioning

confidence: 99%

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Globally generated vector bundles on a smooth quadric surface

2015

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“…and so E admits the following resolution (1). Similarly when deg(Z) = 2, we obtain the resolution (2).…”

Section: Rank Two Globally Generated Vector Bundle Withmentioning

confidence: 72%

Section: Rank Two Globally Generated Vector Bundle Withmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Globally generated vector bundles on a smooth quadric surface

2015

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“…The m-effective pair associated to E is in this case (4, 6). In the recent work [ACM13] it is shown that the only possibilities for such a pair are to be associated to either a quotient of O P 2 (1) 4 , or a quotient of O P 2 (1) ⊕ O P 2 (2) ⊕ Q. The first case corresponds to a stable Steiner bundle, while the second one is semistable.…”

Section: Preliminariesmentioning

confidence: 99%

Planes of matrices of constant rank and globally generated vector bundles

Boralevi¹,

Mezzetti²

2015

Ann. inst. Fourier

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We consider the problem of determining all pairs (c1, c2) of Chern classes of rank 2 bundles that are cokernel of a skew-symmetric matrix of linear forms in 3 variables, having constant rank 2c1 and size 2c1 + 2. We completely solve the problem in the "stable" range, i.e. for pairs with c 2 1 − 4c2 < 0, proving that the additional condition c2 ≤ c 1 +1 2 is necessary and sufficient. For c 2 1 − 4c2 ≥ 0, we prove that there exist globally generated bundles, some even defining an embedding of P 2 in a Grassmannian, that cannot correspond to a matrix of the above type. This extends previous work on c1 ≤ 3.2010 Mathematics Subject Classification. 14J60, 15A30. Key words and phrases. Skew-symmetric matrices, constant rank, globally generated vector bundles. Research partially supported by MIUR funds, PRIN 2010-2011 project "Geometria delle varietà algebriche", and by Università degli Studi di Trieste -FRA 2013 project "Geometria e topologia delle varietà".

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Locally Cohen-Macaulay space curves defined by cubic equations and globally generated vector bundles

Anghel,

Coanda,

Manolache

2015

Preprint

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Let E be a globally generated vector bundle on the projective 3-space, with c 1 at least 4, which is not the quotient of a larger vector bundle by a trivial subbundle. We show that if E(−c 1 +3) has a non-zero global section then, except for some cases that can be listed, E admits as a direct summand a line bundle of degree at least c 1 − 3. This reduces the problem of the classification of globally generated vector bundles with c 1 at most 7 on the projective 3-space to the case where the associated rank 2 reflexive sheaf is stable, a case which requires quite different methods. The proof uses the description of the monads of the ideal sheaves of all locally Cohen-Macaulay space curves defined by cubic equations. We then decide which of the bundles with the above mentioned properties extend to higher dimensional projective spaces. This part uses, among other things, a result of Barth and Ellencwajg [Lecture Notes Math. 683 (1978), Springer Verlag, 1-24] asserting the non-existence of stable rank 2 vector bundles on the projective 4-space with c 1 = 0, c 2 = 3. We provide a different proof of this fact, based on a theorem of Mohan Kumar, Peterson and Rao [Manuscripta Math. 112 (2003), 183-189]. Finally, we recover quickly, using the previously mentioned results, the classification of globally generated vector bundles with c 1 = 4 on the projective n-space for n at least 4, which is part of the main result of our previous paper [arXiv:1305.3464]. We provide, in the appendices to the paper, graded free resolutions for the homogeneous ideals and for the graded structural algebras of all nonreduced locally Cohen-Macaulay space curves of degree at most 4. We use only a small part of the content of the appendices but we decided to give the complete list because it might be useful in some other contexts, too (compare with the papers of Nollet [

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Globally Generated Vector Bundles on P^n with c_1=4

Abstract: We provide a complete classification of globally generated vector bundles with first Chern class c 1 ≤ 5 one the projective plane and with c 1 ≤ 4 on the projective n-space for n ≥ 3. This reproves and extends, in a systematic manner, previous results obtained for c 1

Cited by 3 publications

References 35 publications

Globally generated vector bundles on a smooth quadric surface

Globally generated vector bundles on a smooth quadric surface

Planes of matrices of constant rank and globally generated vector bundles

Locally Cohen-Macaulay space curves defined by cubic equations and globally generated vector bundles

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