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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0005-y,
author = {Edoardo Ballico and Francesco Malaspina and Paolo Valabrega and Mario Valenzano},
title = {On Buchsbaum bundles on quadric hypersurfaces},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1361-1379},
zbl = {1282.14031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0005-y}
}
Edoardo Ballico; Francesco Malaspina; Paolo Valabrega; Mario Valenzano. On Buchsbaum bundles on quadric hypersurfaces. Open Mathematics, Tome 10 (2012) pp. 1361-1379. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0005-y/
[1] Ballico E., Valabrega P., Valenzano M., Non-vanishing theorems for rank two vector bundles on threefolds, Rend. Istit. Mat. Univ. Trieste, 2011, 43, 11–30 | Zbl 1253.14041
[2] Chang M.-C., Characterization of arithmetically Buchsbaum subschemes of codimension 2 in ℙn, J. Differential Geom., 1990, 31(2), 323–341 | Zbl 0663.14034
[3] Ein L., Sols I., Stable vector bundles on quadric hypersurfaces, Nagoya Math. J., 1984, 96, 11–22 | Zbl 0558.14013
[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 | Zbl 0657.14027
[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
[6] Hernández R., Sols I., On a family of rank 3 bundles on Gr(1, 3), J. Reine Angew. Math., 1985, 360, 124–135
[7] Huybrechts D., Lehn M., The Geometry of Moduli Spaces of Sheaves, Aspects Math., Friedrich Vieweg & Sohn, Braunschweig, 1997 | Zbl 0872.14002
[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] 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] 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 | Zbl 0657.14006
[11] Ottaviani G., On Cayley bundles on the five-dimensional quadric, Boll. Un. Mat. Ital. A, 1990, 4(1), 87–100 | Zbl 0722.14006
[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 | Zbl 0839.14016
[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 | Zbl 0578.14014
[14] Valenzano M., Rank 2 reflexive sheaves on a smooth threefold, Rend. Semin. Mat. Univ. Politec. Torino, 2004, 62(3), 235–254 | Zbl 1183.14026