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Curves and vector bundles on quartic threefolds

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2009
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Maddona, Carlo G.
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Korean Mathematical Society
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In this paper we study arithmetically Cohen-Macaulay (ACM for short) vector bundles E of rank k 3 on hypersurfaces Xr P4 of degree r 1. We consider here mainly the case of degree r = 4, which is the first unknown case in literature. Under some natural conditions for the bundle E we derive a list of possible Chern classes (c1, c2, c3) which may arise in the cases of rank k = 3 and k = 4, when r = 4 and we give several examples.
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[1] E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of Algebraic Curves, Springer, 1985. [2] E. Arbarello and E. Sernesi, Petri’s approach to the study of the ideal associated to a special divisor, Inventiones Math. 49(1978), 99–119. [3] E. Arrondo, A home-made Hartshorne-Serre correspondence, Rev.Mat. Complut. 20 (2007), no. 2, 423–443. [4] E. Arrondo and L. Costa, Vector bundles on Fano 3–folds without intermediate cohomology, Comm. Algebra 28 (2000), no. 8, 3899–3911. [5] E. Arrondo and D. Faenzi, Vector bundles with no intermediate cohomology on Fano threefolds of type V22, Pacific J. Math. 225 (2006), no. 2, 201–220. [6] J. Carlson, M. Green, P. Griffiths and J. Harris, Infinitesimal variations of Hodge structures (I), Comp. Math. 50 (1983), 105–205. [7] L. Chiantini and C. Madonna, ACM bundles on a general quintic threefold, Matematiche (Catania) 55(2000), no.2, 239–258. [8] L. Chiantini and C. Madonna, A splitting criterion for rank 2 vector bundles on a general sextic threefold, Internat. J. Math. 15(2004), no.4, 341–359. [9] L. Chiantini and C.Madonna, ACM bundles on general hypersurfaces in P5 of low degree, Collect. Math. Vol. 56(2005), no. 1, 85-96. [10] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer 1999. [11] D. Eisenbud, J. Koh, and M. Stillman, Determinantal Equations for Curves of High Degree Amer. J. Math. Vol. 110(1988), no. 3, pp. 513- 539. [12] D. Faenzi, Bundles over the Fano threefold V5, Comm. Algebra 33(2005), no. 9, 3061–3080. [13] W. Fulton, Intersection theory, Springer 1998. [14] J. Harris, M. Roth and J. Starr, Curves of small degree on cubic threefolds, Rocky Mountain J. Math. 35(2005), no. 3, 761–817. [15] G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. 14 (1964), 689-713. [16] A. Iliev and D. Markushevich, Quartic 3-folds: pfaffians, vector bundles, and half-canonical curves, Mich. Math. J. 47 (2000), 385–394. [17] A. Iliev and L. Manivel, Pfaffian lines and vector bundles on Fano threefolds of genus 8, J. Alg. Geom., to appear. [18] N.M. Kumar, A.P. Rao and G.V. Ravindra, Arithmetically Cohen- Macaulay bundles on three dimensional hypersurfaces, Int. Math. Res. Not. IMRN 2007, no. 8, Art. ID rnm025, 11 pp. [19] C. Madonna, A splitting criterion for rank 2 vector bundles on hypersurfaces in P4, Rend. Sem. Mat. Univ. Pol. Torino 56 (1998), no.2, 43–54. [20] C. Madonna, Rank–two vector bundles on general quartic hypersurfaces in P4, Rev. Mat. Complut. XIII (2000), num.2, 287–301. [21] C.G.Madonna, ACM vector bundles on prime Fano threefolds and complete intersection Calabi Yau threefolds, Rev. Roumaine Math. Pures Appl. 47(2002) no.2, 211-222. [22] C. Madonna, Rank 4 vector bundles on the quintic threefold, CEJM 3 (2005), no.3, 404–411. [23] S. Mori, On degrees and genera of curves on smooth quartic surfaces in P3, Nagoya Math. J. 96(1984), 127–132. [24] B.G. Mo˘ıˇsezon, Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967) 225–268.
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