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Vector bundles with no intermediate cohomology on Fano threefolds of type V-22

Arrondo Esteban, Enrique and Faenzi, Daniele (2006) Vector bundles with no intermediate cohomology on Fano threefolds of type V-22. Pacific Journal of Mathematics, 225 (2). pp. 201-220. ISSN 0030-8730

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We classify rank-2 vector bundles with no intermediate cohomology on the general prime Fano threefold of index 1 and genus 12. The structure of their moduli spaces is given by means of a monad-theoretic resolution in terms of exceptional bundles.

Item Type:Article
Uncontrolled Keywords:Cohen-Macaulay modules; intermediate cohomology; Fano threefolds V-22; canonical and half-canonical curves
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:14807

[Arrondo and Costa 2000] E. Arrondo and L. Costa, “Vector bundles on Fano 3-folds without intermediate

cohomology”, Comm. Algebra 28:8 (2000), 3899–3911. MR2001d:14043 Zbl 1004.14010

[Arrondo and Graña 1999] E. Arrondo and B. Graña, “Vector bundles on G(1, 4) without intermediate

cohomology”, J. Algebra 214:1 (1999), 128–142. MR 2000e:14069 Zbl 0963.14027

[Buchweitz et al. 1987] R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer, “Cohen–Macaulay

modules on hypersurface singularities. II”, Invent. Math. 88:1 (1987), 165–182. MR 88d:14005

Zbl 0617.14034

[Drezet 1986] J.-M. Drezet, “Fibrés exceptionnels et suite spectrale de Beilinson généralisée sur

P2(C)”, Math. Ann. 275:1 (1986), 25–48. MR 88b:14014 Zbl 0578.14013

[Ellingsrud et al. 1987] G. Ellingsrud, R. Piene, and S. A. Strømme, “On the variety of nets of

quadrics defining twisted cubics”, pp. 84–96 in Space curves (Rocca di Papa, 1985), Lecture Notes

in Math. 1266, Springer, Berlin, 1987. MR 88h:14034 Zbl 0659.14027

[Faenzi 2005] D. Faenzi, “Bundles over the Fano threefold V5”, Comm. Algebra 33:9 (2005), 3061–

3080. MR 2006g:14067 Zbl 1082.14044

[Faenzi 2006] D. Faenzi, “Bundles over Fano threefolds of type V22”, Ann. Mat. Pura Appl. (2006).

To appear.

[Furushima and Nakayama 1989] M. Furushima and N. Nakayama, “The family of lines on the Fano

threefold V5”, Nagoya Math. J. 116 (1989), 111–122. MR 90k:14039 Zbl 0731.14025

[Gorodentsev 1990] A. L. Gorodentsev, “Exceptional objects and mutations in derived categories”,

pp. 57–73 in Helices and vector bundles, London Math. Soc. Lecture Note Ser. 148, Cambridge

Univ. Press, Cambridge, 1990. MR 91m:14024 Zbl 0722.14007

[Hartshorne 1974] R. Hartshorne, “Varieties of small codimension in projective space”, Bull. Amer.

Math. Soc. 80 (1974), 1017–1032. MR 52 #5688 Zbl 0304.14005

[Horrocks 1964] G. Horrocks, “Vector bundles on the punctured spectrum of a local ring”, Proc.

London Math. Soc. (3) 14 (1964), 689–713. MR 30 #120 Zbl 0126.16801

[Huybrechts and Lehn 1997] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves,

Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 98g:14012

Zbl 0872.14002

[Iskovskih 1978] V. A. Iskovskih, “Fano threefolds, II”, Izv. Akad. Nauk SSSR Ser. Mat. 42:3 (1978),

506–549. In Russian; translated in Math. USSR Izv. 12:3 (1978), 469–506. MR 80c:14023b

[Iskovskikh 1989] V. A. Iskovskikh, “Double projection from a line onto Fano 3-folds of the first

kind”, Mat. Sb. 180:2 (1989), 260–278. In Russian; translated in Math. USSR Sb. 66:1 (1990),

265–284. MR 90j:14048 Zbl 0691.14027

[Iskovskikh and Prokhorov 1999] V. A. Iskovskikh and Y. G. Prokhorov, “Fano varieties”, pp.

1–247 in Algebraic geometry, vol. V, Encyclopaedia Math. Sciences 47, Springer, Berlin, 1999.

MR 2000b:14051b Zbl 0912.14013

[Katz and Strømme 1992] S. Katz and S. A. Strømme, “Schubert, a Maple package for intersection

theory and enumerative geometry”, 1992, Available at

[Knörrer 1987] H. Knörrer, “Cohen–Macaulay modules on hypersurface singularities, I”, Invent.

Math. 88:1 (1987), 153–164. MR 88d:14004 Zbl 0617.14033

[Kuznetsov 1996] A. G. Kuznetsov, “An exception set of vector bundles on the varieties V22”, Vestnik

Moskov. Univ. Ser. I Mat. Mekh. 3 (1996), 41–44, 92. In Russian; translation in Moscow Univ.

Math. Bull. 51:3 (1996), 35–37. MR 97m:14040 Zbl 0913.14010

[Madonna 2000] C. Madonna, “Rank-two vector bundles on general quartic hypersurfaces in P4”,

Rev. Mat. Complut. 13:2 (2000), 287–301. MR 2002d:14064 Zbl 0981.14019

[Madonna 2002] C. G. Madonna, “ACM vector bundles on prime Fano threefolds and complete intersection

Calabi–Yau threefolds”, Rev. Roumaine Math. Pures Appl. 47:2 (2002), 211–222 (2003).

MR 2004g:14048 Zbl 1051.14050

[Maruyama 1981] M. Maruyama, “The theorem of Grauert–Mülich–Spindler”, Math. Ann. 255:3

(1981), 317–333. MR 82k:14012 Zbl 0438.14015

[Mukai 1992] S. Mukai, “Fano 3-folds”, pp. 255–263 in Complex projective geometry (Trieste and

1989 / Bergen, 1989), edited by G. Ellingsrud et al., London Math. Soc. Lecture Note Ser. 179,

Cambridge Univ. Press, Cambridge, 1992. MR 94a:14042 Zbl 0774.14037

[Mukai 2004] S. Mukai, “Plane quartics and Fano threefolds of genus twelve”, pp. 563–572 in The

Fano Conference (Torino, 2002), edited by A. Collino et al., Università di Torino, Dipartimento

di Matematica, Turin, 2004, Available at MR 2006d:14043 Zbl 1068.14050

[Okonek et al. 1980] C. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective

spaces, Progress in Mathematics 3, Birkhäuser, Mass., 1980. MR 81b:14001 Zbl 0438.32016

[Ottaviani 1987] G. Ottaviani, “Critères de scindage pour les fibrés vectoriels sur les Grassmanniennes

et les quadriques”, C. R. Acad. Sci. Paris Sér. I Math. 305:6 (1987), 257–260. MR 88j:14024

Zbl 0629.14012

[Ottaviani 1989] G. Ottaviani, “Some extensions of Horrocks criterion to vector bundles on Grassmannians

and quadrics”, Ann. Matematica Pura Appl. (4) 155 (1989), 317–341. MR 91f:14018

Zbl 0718.14010

[Rudakov 1990] A. N. Rudakov, “Exceptional collections, mutations and helices”, pp. 1–6 in Helices

and vector bundles, London Math. Soc. Lecture Note Ser. 148, Cambridge Univ. Press, Cambridge,

1990. Seminaire Rudakov, Translated from the Russian by A. D. King, P. Kobak and A. Maciocia.

MR 93a:14016 Zbl 0721.14011

[Schreyer 2001] F.-O. Schreyer, “Geometry and algebra of prime Fano 3-folds of genus 12”, Compositio

Math. 127:3 (2001), 297–319. MR 2002d:14062 Zbl 1049.14036

[Serre 1963] J.-P. Serre, “Sur les modules projectifs”, in Séminaire d’algèbre et de théorie de nombres

14 (Séminaire P. Dubreil), 1960/1961, Sécrétariat mathématique, Paris, 1963. Exp. no. 2.

Zbl 0132.41301

[Takeuchi 1989] K. Takeuchi, “Some birational maps of Fano 3-folds”, Compositio Math. 71:3

(1989), 265–283. MR 90k:14041 Zbl 0712.14025

[Vogelaar 1978] J. A. Vogelaar, Constructing vector bundles from codimension-two subvarieties Ph.D. thesis, Univ. Leiden, 1978

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