On Buchsbaum bundles on quadric hypersurfaces

Edoardo Ballico; Francesco Malaspina; Paolo Valabrega; Mario Valenzano

Open Mathematics (2012)

  • Volume: 10, Issue: 4, page 1361-1379
  • ISSN: 2391-5455

Abstract

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Let E be an indecomposable rank two vector bundle on the projective space ℙn, n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q n ⊂ ℙn+1, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q 3, k ≥ 2, we prove two boundedness results.

How to cite

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Edoardo Ballico, et al. "On Buchsbaum bundles on quadric hypersurfaces." Open Mathematics 10.4 (2012): 1361-1379. <http://eudml.org/doc/269458>.

@article{EdoardoBallico2012,
abstract = {Let E be an indecomposable rank two vector bundle on the projective space ℙn, n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q n ⊂ ℙn+1, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q 3, k ≥ 2, we prove two boundedness results.},
author = {Edoardo Ballico, Francesco Malaspina, Paolo Valabrega, Mario Valenzano},
journal = {Open Mathematics},
keywords = {Arithmetically Buchsbaum rank two vector bundles; Smooth quadric hypersurfaces; arithmetically Buchsbaum rank two vector bundles; smooth quadric hypersurfaces},
language = {eng},
number = {4},
pages = {1361-1379},
title = {On Buchsbaum bundles on quadric hypersurfaces},
url = {http://eudml.org/doc/269458},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Edoardo Ballico
AU - Francesco Malaspina
AU - Paolo Valabrega
AU - Mario Valenzano
TI - On Buchsbaum bundles on quadric hypersurfaces
JO - Open Mathematics
PY - 2012
VL - 10
IS - 4
SP - 1361
EP - 1379
AB - Let E be an indecomposable rank two vector bundle on the projective space ℙn, n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q n ⊂ ℙn+1, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q 3, k ≥ 2, we prove two boundedness results.
LA - eng
KW - Arithmetically Buchsbaum rank two vector bundles; Smooth quadric hypersurfaces; arithmetically Buchsbaum rank two vector bundles; smooth quadric hypersurfaces
UR - http://eudml.org/doc/269458
ER -

References

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  2. [2] Chang M.-C., Characterization of arithmetically Buchsbaum subschemes of codimension 2 in ℙn, J. Differential Geom., 1990, 31(2), 323–341 Zbl0663.14034
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  4. [4] Ellia Ph., Fiorentini M., Quelques remarques sur les courbes arithmétiquement Buchsbaum de l’espace projectif, Ann. Univ. Ferrara Sez. VII, 1987, 33, 89–111 Zbl0657.14027
  5. [5] Ellia Ph., Sarti A., On codimension two k-Buchsbaum subvarieties of ℙn, In: Commutative Algebra and Algebraic Geometry, Ferrara, Lecture Notes in Pure and Appl. Math., 206, Marcel Dekker, New York, 1999, 81–92 
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  7. [7] Huybrechts D., Lehn M., The Geometry of Moduli Spaces of Sheaves, Aspects Math., Friedrich Vieweg & Sohn, Braunschweig, 1997 Zbl0872.14002
  8. [8] Madonna C., A splitting criterion for rank 2 vector bundles on hypersurfaces in ℙ4, Rend. Semin. Mat. Univ. Politec. Torino, 1998, 56(2), 43–54 
  9. [9] Kumar N.M., Rao A.P., Buchsbaum bundles on ℙn, J. Pure Appl. Algebra, 2000, 152(1–3), 195–199 http://dx.doi.org/10.1016/S0022-4049(99)00129-2 
  10. [10] Ottaviani G., Spinor bundles on quadrics, Trans. Amer. Math. Soc., 1988, 307(1), 301–316 http://dx.doi.org/10.1090/S0002-9947-1988-0936818-5 Zbl0657.14006
  11. [11] Ottaviani G., On Cayley bundles on the five-dimensional quadric, Boll. Un. Mat. Ital. A, 1990, 4(1), 87–100 Zbl0722.14006
  12. [12] Ottaviani G., Szurek M., On moduli of stable 2-bundles with small Chern classes on Q 3, Ann. Mat. Pura Appl., 1994, 167(1), 191–241 http://dx.doi.org/10.1007/BF01760334 Zbl0839.14016
  13. [13] Sols I., On spinor bundles, J. Pure Appl. Algebra, 1985, 35(1), 85–94 http://dx.doi.org/10.1016/0022-4049(85)90031-3 Zbl0578.14014
  14. [14] Valenzano M., Rank 2 reflexive sheaves on a smooth threefold, Rend. Semin. Mat. Univ. Politec. Torino, 2004, 62(3), 235–254 Zbl1183.14026

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