Abstract
We extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines. We introduce a set of invariants for a vector bundle on the product of two projective lines, which includes the first cohomology module of the bundle, and prove that there is a one to one correspondence between these sets of invariants and isomorphism classes of vector bundles without line bundle summands.
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Arrondo, E., Costa, L.: Vector bundles on Fano 3-folds without intermediate cohomology. Commun. Algebra 28(8), 3899–3911 (2000)
Arrondo, E., Faenzi, D.: Vector bundles with no intermediate cohomology on Fano threefolds of type \(V_{22}\). Pac. J. Math. 225(2), 201–220 (2006)
Arrondo, E., Graña, B.: Vector bundles on \(G(1,4)\) without intermediate cohomology. J. Algebra 214, 128–142 (1999)
Arrondo, E., Malaspina, F.: Cohomological characterization of vector bundles on Grassmannians of lines. J. Algebra 323, 1098–1106 (2010)
Bakhtary, P.: Splitting criteria for vector bundles on higher-dimensional varieties. Pac. J. Math. 252(1), 19–30 (2011)
Ballico, E., Malaspina, F.: Qregularity and an extension of the Evans–Griffiths criterion to vector bundles on quadrics. J. Pure Appl. Algebra 213(2), 194–202 (2009)
Ballico, E., Malaspina, F.: Regularity and cohomological splitting conditions for vector bundles on multiprojectives spaces. J. Algebra 345, 137–149 (2011)
Barth, W., Hulek, K.: Monads and moduli of vector bundles. Manuscripta Math. 25, 323–447 (1978)
Biswas, J., Ravindra, G.V.: Arithmetically Cohen–Macaulay bundles on complete intersection varieties of sufficiently high multi-degree. Math. Z. 265(3), 493–509 (2010)
Brambilla, C., Faenzi, D.: Moduli space of rank 2 ACM bundles on prime Fano threefolds. Mich. Math. J. 60(1), 113–148 (2011)
Buchweitz, R.-O., Greuel, G.-M., Schreyer, F.-O.: Cohen-Macaulay modules on hypersurface singularities II. Invent. Math. 88, 165–182 (1987)
Buraggina, A.: The intermediate cohomology modules of a rank two vector bundle on \({ P}^3\) are not split. Commun. Algebra 25(3), 999–1008 (1997)
Casanellas, M., Hartshorne, R.: ACM bundles on cubic surfaces. J. Eur. Math. Soc. 13(3), 709–731 (2011)
Casanellas, M., Hartshorne, R., Geiss, F., Schreyer, F.-O.: Stable Ulrich bundles. Int. J. Math. 23(8), 1250083 (2012)
Chiantini, L., Faenzi, D.: Rank 2 arithmetically Cohen-Macaulay bundles on a general quintic surface. Math. Nachr. 282(12), 1691–1708 (2009)
Chiantini, L., Madonna, C.: ACM bundles on a general quintic threefold. Dedicated to Silvio Greco on the occasion of his 60th birthday (Catania 2001). Matematiche 55(2), 239–258 (2000)
Chiantini, L., Madonna, C.: A splitting criterion for rank 2 bundles on a general sextic threefold. Int. J. Math. 15(4), 341–359 (2004)
Chiantini, L., Madonna, C.: ACM bundles on general hypersurfaces in \({ P}^5\) of low degree. Collect. Math. 56(1), 85–96 (2005)
Chiantini, L., Valabrega, P.: Subcanonical curves and complete intesections in the projective 3-space. Ann. Mat. Pura Appl. 138, 309–330 (1984)
Costa, L., Miró-Roig, R.M.: Cohomological characterization of vector bundles on multiprojective spaces. J. Algebra 294(1), 73–96 (2005), with a corrigendum 319(3), 1336–1338 (2008)
Costa, L., Miró-Roig, R.M.: Homogeneous Ulrich bundles on Grassmannians of lines (2012)
Ein, L.: An analogue of Max Noether’s theorem. Duke Math. J. 52(3), 689–706 (1985)
Evans, E.G., Griffiths, P.: The syzygy problem. Ann. Math. 114(2), 323–333 (1981)
Faenzi, D.: Bundles over the Fano threefold \(V_5\). Commun. Algebra 33(9), 3061–3080 (2005)
Faenzi, D.: Rank 2 arithmetically Cohen-Macaulay bundles on a nonsingular cubic surface. J. Algebra 319, 143–186 (2008)
Horrocks, G.: Vector bundles on the punctured spectrum of a ring. Proc. Lond. Math. Soc. (3) 14, 689–713 (1964)
Knörrer, H.: Cohen-Macaulay modules of hypersurface singularities I. Invent. Math. 88, 153–164 (1987)
Le Potier, J.: Sur l’espace de modules des fibrés de Yang et Mills. Mathematics and physics (Paris, 1979/1982). Prog. Math. 37, 65–137. Birkhäuser, Boston (1983)
Maruyama, M.: Stable vector bundles on an algebraic surface. Nagoya Math. J. 58, 25–68 (1975)
Mohan Kumar, N., Peterson, C., Rao, A.P.: Monads on projective spaces. Manuscripta Math. 112, 183–189 (2003)
Mohan Kumar, N., Rao, A.P., Ravindra, G.V.: Arithmetically Cohen-Macaulay bundles on hypersurfaces. Comment. Math. Helv. 82(4), 829–843 (2007)
Mohan Kumar, N., Rao, A.P., Ravindra, G.V.: Arithmetically Cohen-Macaulay bundles on three dimensional hypersurfaces. Int. Math. Res. Not. no. 8, Art. ID rnm025, 11 pp (2007)
Mohan Kumar, N., Rao, A.P., Ravindra, G.V.: On codimension two subvarieties in hypersurfaces. In: de Jue, Rob, et al. (eds.), Motives and algebraic cycles. pp. 167–174, Fields Institute Communications, 56, American Mathematical Society, Providence, RI (2009)
Ottaviani, G.: Spinor bundles on quadrics. Trans. Am. Math. Soc. 307(1), 301–316 (1988)
Ottaviani, G.: Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics. Ann. Mat. Pura Appl. (4) 155, 317–341 (1989)
Rao, A.P.: A note on cohomology modules of rank two bundles. J. Algebra 86, 23–34 (1984)
Ravindra, G.V.: Curves on threefolds and a conjecture of Griffiths-Harris. Math. Ann. 345(3), 731–748 (2009)
Soberon-Chavez, S.: Rank 2 vector bundles over a complex quadric surface. Q. J. Math. Oxf. Ser. (2) 36, 159–172 (1985)
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F. Malaspina: Partially supported by GNSAGA of Indam (Italy).
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Malaspina, F., Rao, A.P. Horrocks Correspondence on a Quadric Surface. Geom Dedicata 169, 15–31 (2014). https://doi.org/10.1007/s10711-013-9839-0
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DOI: https://doi.org/10.1007/s10711-013-9839-0