We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail.
@article{bwmeta1.element.doi-10_2478_s11533-012-0055-1,
author = {Alexander Kuznetsov},
title = {Instanton bundles on Fano threefolds},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1198-1231},
zbl = {1282.14075},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0055-1}
}
Alexander Kuznetsov. Instanton bundles on Fano threefolds. Open Mathematics, Tome 10 (2012) pp. 1198-1231. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0055-1/
[1] Atiyah M.F., Hitchin N.J., Drinfel’d V.G., Manin Yu.I., Construction of instantons, Phys. Lett. A, 1978, 65, 185–187 http://dx.doi.org/10.1016/0375-9601(78)90141-X
[2] Bernardara M., Macrì E., Mehrotra S., Stellari P., A categorical invariant for cubic threefolds, Adv. Math., 2012, 229(2), 770–803 http://dx.doi.org/10.1016/j.aim.2011.10.007 | Zbl 1242.14012
[3] Bondal A.I., Representations of associative algebras and coherent sheaves, Math. USSR-Izv., 1990, 34(1), 23–42 http://dx.doi.org/10.1070/IM1990v034n01ABEH000583 | Zbl 0692.18002
[4] Bondal A.I., Kapranov M.M., Math. USSR-Izv., 1990, 35(3), 519–541 http://dx.doi.org/10.1070/IM1990v035n03ABEH000716
[5] Bondal A., Orlov D., Semiorthogonal decomposition for algebraic varieties, preprint available at http://arxiv.org/abs/alg-geom/9506012
[6] Bondal A., Orlov D., Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math., 2011, 125(3), 327–344 http://dx.doi.org/10.1023/A:1002470302976 | Zbl 0994.18007
[7] Faenzi D., Even and odd instanton bundles on Fano threefolds of Picard number one, preprint available at http://arxiv.org/abs/1109.3858
[8] Fujita T., On the structure of polarized varieties with Δ-genera zero, J. Fac. Sci. Univ Tokyo. Sec. IA Math., 1975, 22, 103–115 | Zbl 0333.14004
[9] Hoppe H.J., Generischer Spaltungstyp und zweite Chernklasse stabiler Vektorraumbündel vom Rang 4 auf ℙ4, Math. Z., 1984, 187(3), 345–360 http://dx.doi.org/10.1007/BF01161952 | Zbl 0567.14011
[10] Ingalls C., Kuznetsov A., On nodal Enriques surfaces and quartic double solids, preprint available at http://arxiv.org/abs/1012.3530 | Zbl 06399403
[11] Iskovskikh V.A., Prokhorov Yu.G., Fano Varieties, Algebraic Geometry, 5, Encyclopaedia Math. Sci., 47, Springer, Berlin, 1999
[12] Kapustin A., Kuznetsov A., Orlov D., Noncommutative instantons and twistor transform, Comm. Math. Phys., 2001, 221(2), 385–432 http://dx.doi.org/10.1007/PL00005576 | Zbl 0989.81127
[13] Kuznetsov A., Derived category of a cubic threefold and the variety V 14, Proc. Steklov Inst. Math., 2004, 246(3), 171–194
[14] Kuznetsov A., Derived categories of quadric fibrations and intersections of quadrics, Adv. Math., 2008, 218(5), 1340–1369 http://dx.doi.org/10.1016/j.aim.2008.03.007 | Zbl 1168.14012
[15] Kuznetsov A.G., Derived categories of Fano threefolds, Proc. Steklov Inst. Math, 2009, 264(1), 110–122 http://dx.doi.org/10.1134/S0081543809010143 | Zbl 1312.14055
[16] Markushevich D., Tikhomirov A., The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geom., 2001, 10(1), 37–62 | Zbl 0987.14028
[17] Mukai S., Duality between D(X) and with its application to Picard sheaves, Nagoya Math. J., 1981, 81, 153–175
[18] Okonek C., Schneider M., Spindler H., Vector Bundles on Complex Projective Spaces, Progr. Math., 3, Birkhaüser, Boston, 1980 http://dx.doi.org/10.1007/978-3-0348-0151-5 | Zbl 0438.32016
[19] Okonek C., Spindler H., Mathematical instanton bundles on ℙ2n+1, J. Reine Angew. Math., 1986, 364, 35–50 | Zbl 0568.14009
[20] Orlov D., Exceptional set of vector bundles on the variety V 5, Moscow Univ. Math. Bull., 1991, 46(5), 48–50
[21] Orlov D., Derived categories of coherent sheaves and triangulated categories of singularities, In: Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin, 2, Progr. Math., 270, Birkhaüser, Boston, 2009, 503–531 | Zbl 1200.18007
[22] Popa M., Generalized theta linear series on moduli spaces of vector bundles on curves, In: Handbook of Moduli (in press) | Zbl 1322.14015
[23] Raynaud M., Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France, 1982, 110(1), 103–125 | Zbl 0505.14011
[24] Spindler H., Trautmann G., Special instanton bundles on ℙ2N+1, their geometry and their moduli, Math. Ann., 1990, 286(1–3), 559–592 http://dx.doi.org/10.1007/BF01453589 | Zbl 0752.14014
[25] Wall C.T.C., Nets of quadrics, and theta-characteristics of singular curves, Philos. Trans. Roy. Soc. London Ser. A, 1978, 289(1357), 229–269 http://dx.doi.org/10.1098/rsta.1978.0060