The stability of certain vector bundles on ℙ n

Bohnhorst, Guntram; Spindler, Heinz

doi:10.1007/bfb0094509

Cited by 40 publications

(61 citation statements)

References 11 publications

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“…By proposition 1.4 of [4] and by the fact that −2γ < −γ − nα, the last sequence is the minimal resolution of F α,γ (γ): hence Q α,γ (−2γ) = Coker φ is directly defined by this resolution. is induced by an isomorphism of sequences:…”

Section: Small Deformations Of F αγmentioning

confidence: 95%

Weighted Tango bundles on Pn and their moduli spaces

Cascini¹

2001

Forum Mathematicum

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We de®ne a new class of algebraic n À 1-bundles on P n , that contains the bundles introduced by Tango [14] and their stable generalized pull-backs; we show that these bundles are invariant under small deformations and that they correspond to smooth points of moduli spaces.1991 Mathematics Subject Classi®cation: 14F05.It is a very di½cult problem to ®nd examples of non-splitting algebraic vector bundles on the complex projective space P n whose rank is less than n. In particular for n 6 the only known examples are essentially the mathematical instantons [3] (for odd n) and the bundles introduced by Tango [14]: all of them have rank n À 1. Of course, pulling back the Tango bundles by a ®nite morphism P n 3 P n gives other examples of rank n À 1 bundles.In [9], Horrocks introduced a new technique of constructing new bundles from old ones, which generalizes the pull-back. This method, that we can call generalized pullback, has been extensively studied in [1] and [2] and it applies only to bundles whose symmetry group contains a copy of C Ã .In this paper we show that, for any n 3, there exists a Tango bundle that is SL2-invariant: hence the generalized pull-back allows us to de®ne a new class of n À 1-bundles on P n .More precisely, let Y g be integer numbers such that g b n 0 and let Q Y g be the bundles on P n described by the exact sequence:Q Y g can also be de®ned as the generalized pull-back of the quotient bundle on P n and, in particular, Q 0Y 1 is the quotient bundle. Let us de®ne the rank 2n À 1 vector bundle:O P n 2n À 1 À 2kXBrought to you by |

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Section: Small Deformations Of F αγmentioning

confidence: 95%

Weighted Tango bundles on Pn and their moduli spaces

Cascini¹

2001

Forum Mathematicum

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“…In Section 1, we classify all Betti numbers of rank r bundles on P n satisfying ( †), generalizing results from Bohnhorst and Spindler [3] for the case r = n. Accordingly, we classify all possible Hilbert functions of such bundles, and introduce a compact way to represent and to generate them. We show that there are only finitely many possible Betti numbers of bundles satisfying ( †) with fixed first Chern class and bounded regularity, generalizing the observation of Dionisi and Maggesi [4] for r = n = 2.…”

Section: Introductionmentioning

confidence: 99%

Bundles on with vanishing lower cohomologies

Zhang

2020

Can. J. Math.-J. Can. Math.

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We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We classify the Betti numbers of these bundles and show that there are only finitely many possibilities with a given first Chern class and bounded regularity. Accordingly, we classify the Hilbert functions of these bundles and provide an efficient way to represent and to generate them. Finally, we show that the Betti numbers of bundles with a fixed Hilbert function form a graded lattice. We then describe the stratification of the space of isomorphism classes of bundles with a fixed Hilbert function by Betti numbers.

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“…They were introduced by Dolgachev and Kapranov in the context of hyperplane arrangements on P n (cf. [14]), and have been studied by several other authors since then, see for instance [2,6,7] for Steiner bundles on projective spaces and [26] for Steiner bundles on quadrics. Here we apply the same technique we used to characterize linear bundles, once again without using Beilinson spectral sequence, in order to obtain a characterization of Steiner bundles on P n (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Let us now establish the converse of Proposition 6.1 to smooth quadric hypersurfaces, following [6]. We will require the following technical result.…”

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confidence: 98%