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On Buchsbaum bundles on quadric hypersurfaces. (English) Zbl 1282.14031
If $$E$$ is a vector bundle on a polarized projective variety $$(X,L)$$, it is called arithmetically Cohen-Macaulay if all its intermediate cohomologies vanish for all twists of $$L$$. It is called arithmetically Buchsbaum, if the intermediate cohomologies have trivial vector space structure over the ring $$\bigoplus H^0(L^n)$$ and it is properly arithmetically Buchsbaum, if it is in addition not arithmetically Cohen-Macaulay. It is well known that if the variety is $$(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(1))$$, and $$E$$ is indecomposable rank two vector bundle in characteristic zero (so it is properly arithmetically Buchsbaum, by a theorem of Horrocks), then $$n=3$$ and $$E$$ is the null-correlation bundle. In the paper under review, the authors completely classify properly arithmetically Buchsbaum bundles of rank 2 on smooth quadrics $$Q_n$$ in $$\mathbb{P}^{n+1}$$

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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##### References:
 [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 · Zbl 0960.14026 [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 · Zbl 0957.14012 [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 · Zbl 0971.14016 [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
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