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On Higher Rank Globally Generated Vector Bundles over a Smooth Quadric Threefold

Part of: (Co)homology theory Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  10 June 2015

E. Ballico ,
S. Huh  and
F. Malaspina
E. Ballico
Affiliation:
Università di Trento, 38123 Povo, Trentino, Italy, ([email protected])
S. Huh
Affiliation:
Department of Mathematics, Sungkyunkwan University, Cheoncheon-dong, Jangan-gu, Suwon 440-746, Republic of Korea, ([email protected])
F. Malaspina
Affiliation:
Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, ([email protected])
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Abstract

We give a complete classification of globally generated vector bundles of rank 3 on a smooth quadric threefold with c1 ≤ 2 and extend the result to arbitrary higher rank case. We also investigate the existence of globally generated indecomposable vector bundles, and give the sufficient and necessary conditions on numeric data of vector bundles for indecomposability.

Keywords

higher rank vector bundlesglobally generatedsmooth quadric threefold

MSC classification

Primary: 14F99: None of the above, but in this section
Secondary: 14J99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 311 - 337
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

1. Ancona, V. and Ottaviani, G., Some applications of Beilinson’s theorem to projective spaces and quadrics, Forum Math. 3(2) (1991), 157–176.Google Scholar
2. Anghel, C. and Manolache, N., Globally generated vector bundles on ℙ n with c 1 = 3, Math. Nachr. 286(14–15) (2013), 1407–1423.CrossRefGoogle Scholar
3. Arrondo, E., A home-made Hartshorne-Serre correspondence, Rev. Mat. Complut. 20(2) (2007), 423–443.CrossRefGoogle Scholar
4. Ballico, E., Huh, S. and Malaspinaü, F., Globally generated vector bundles of rank 2 on a smooth quadric threefold, J. Pure Appl. Alg. 218(2) (2014), 197–207.Google Scholar
5. Ballico, E., Huh, S. and Malaspina, F., Reflexive and spanned sheaves on ℙ3 , Results Math. 65(1–2) (2014), 27–47.CrossRefGoogle Scholar
6. Bănică, C. and Coandă, J., Existence of rank 3 vector bundles with given Chern classes on homogeneous rational 3-folds, Manuscr. Math. 51(1–3) (1985), 121–143.Google Scholar
7. Chang, M.-C., A filtered Bertini-type theorem, J. Reine Angew. Math. 397 (1989), 214–219.Google Scholar
8. Chiodera, L. and Ellia, P., Rank two globally generated vector bundles with c 1 ≤ 5, Rend. Istit. Mat. Univ. Trieste 44 (2012), 413–422.Google Scholar
9. Demazure, M., Pinkham, H. C. and Teissier, B. (eds), Séminaire sur les singularités des surfaces, Lecture Notes in Mathematics, Volume 777 (Springer, 1980).Google Scholar
10. Ellia, P., Chern classes of rank two globally generated vector bundles on ℙ2 , Rend. Lincei Mat. Appl. 24(2) (2013), 147–163.Google Scholar
11. Fujita, T., Defining equations for certain types of polarized varieties, in Complex analysis and algebraic geometry, pp. 165–173 (Iwanami Shoten, Tokyo, 1977).CrossRefGoogle Scholar
12. Green, M. and Lazarsfeld, R., Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67(3) (1988), 301–314.Google Scholar
13. Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, Volume 52 (Springer, 1977).CrossRefGoogle Scholar
14. Hartshorne, R., Stable reflexive sheaves, Math. Annalen 254(2) (1980), 121–176.Google Scholar
15. Hartshorne, R., Genre de courbes algébriques dans l’espace projectif (d’après Gruson, L. et Peskine, C.), Seminaire Bourbaki: volume 1981/1982, Astérisque, Volume 92, pp. 301–313 (Société Mathematique de France, Paris, 1982) (in French).Google Scholar
16. Hartshorne, R. and Hirschowitz, A., Smoothing algebraic space curves, Algebraic geometry, Sitges (Barcelona), 1983, Lecture Notes in Mathematics, Volume 1124, pp. 98–131 (Springer, 1985).Google Scholar
17. Huh, S., On triple Veronese embeddings of ℙ n in the Grassmannians, Math. Nachr. 284(11–12) (2011), 1453–1461.Google Scholar
18. McCleary, J., User’s guide to spectral sequences, Mathematics Lecture Series, Volume 12 (Publish or Perish, Boston, MA, 1985).Google Scholar
19. Manolache, N., Globally generated vector bundles on ℙ3 with c 1 = 3, preprint (arxiv.org/abs/1202.5988, 2012).Google Scholar
20. Okonek, C., Schneider, M. and Spindler, H., Vector bundles on complex projective spaces, Progress in Mathematics, Volume 3 (Birkhäuser, 1980).Google Scholar
21. Ottaviani, G. and Szurek, M., On moduli of stable 2-bundles with small Chern classes on Q 3 , Annali Mat. Pura Applic. (IV) 167 (1994), 191–241.CrossRefGoogle Scholar
22. Sierra, J.-C., A degree bound for globally generated vector bundles, Math. Z. 262(3) (2009), 517–525.Google Scholar
23. Sierra, J.-C. and Ugaglia, L., On globally generated vector bundles on projective spaces, J. Pure Appl. Alg. 213(11) (2009), 2141–2146.Google Scholar
24. Sierra, J.-C. and Ugaglia, L., On globally generated vector bundles on projective spaces II, J. Pure Appl. Alg. 218(1) (2014), 174–180.Google Scholar