Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CURVES AND VECTOR BUNDLES ON QUARTIC THREEFOLDS

  • Arrondo, Enrique (DEPARTAMENTO DE ALGEBRA FACULTAD DE CIENCIAS MATEMATICAS UNIVERSIDAD COMPLUTENSE DE MADRID) ;
  • Madonna, Carlo G. (DEPARTAMENTO DE ALGEBRA FACULTAD DE CIENCIAS MATEMATICAS UNIVERSIDAD COMPLUTENSE DE MADRID)
  • Published : 2009.05.01
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Abstract

In this paper we study arithmetically Cohen-Macaulay (ACM for short) vector bundles $\varepsilon$ of rank k $\geq$ 3 on hypersurfaces $X_r\;{\subset}\;{\mathbb{P}}^4$ of degree r $\geq$ 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 $\varepsilon$ we derive a list of possible Chern classes ($c_1$, $c_2$, $c_3$) 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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References

  1. E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves. Vol. I, Springer-Verlag, New York, 1985
  2. E. Arbarello and E. Sernesi, Petri's approach to the study of the ideal associated to a special divisor, Invent. Math. 49 (1978), no. 2, 99–119 https://doi.org/10.1007/BF01403081
  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 https://doi.org/10.1080/00927870008827064
  5. E. Arrondo and D. Faenzi, Vector bundles with no intermediate cohomology on Fano threefolds of type $V_{22}$, Pacific J. Math. 225 (2006), no. 2, 201–220
  6. J. Carlson, M. Green, P. Griffiths, and J. Harris, Infinitesimal variations of Hodge structure. I, Compositio Math. 50 (1983), no. 2-3, 109–205
  7. L. Chiantini and C. Madonna, A splitting criterion for rank 2 bundles on a general sextic threefold, Internat. J. Math. 15 (2004), no. 4, 341–359
  8. L. Chiantini and C. Madonna, ACM bundles on general hypersurfaces in $P^5$ of low degree, Collect. Math. 56 (2005), no. 1, 85–96
  9. D. Eisenbud, J. Koh, and M. Stillman, Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), no. 3, 513–539
  10. D. Faenzi, Bundles over the Fano threefold $V_5$, Comm. Algebra 33 (2005), no. 9, 3061–3080 https://doi.org/10.1081/AGB-200066116
  11. G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964), 689–713 https://doi.org/10.1112/plms/s3-14.4.689
  12. A. Iliev and D. Markushevich, Quartic 3-fold: Pfaffians, vector bundles, and halfcanonical curves, Michigan Math. J. 47 (2000), no. 2, 385–394 https://doi.org/10.1307/mmj/1030132542
  13. 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
  14. C. Madonna, A splitting criterion for rank 2 vector bundles on hypersurfaces in $P^4$, Rend. Sem. Mat. Univ. Politec. Torino 56 (1998), no. 2, 43–54 (2000)
  15. C. Madonna, Rank-two vector bundles on general quartic hypersurfaces in P^4$, Rev. Mat. Complut. 13 (2000), no. 2, 287–301
  16. C. Madonna, ACM vector bundles on prime Fano threefolds and complete intersection Calabi-Yau threefolds, Rev. Roumaine Math. Pures Appl. 47 (2002), no. 2, 211–222
  17. C. Madonna, Rank 4 vector bundles on the quintic threefold, Cent. Eur. J. Math. 3 (2005), no. 3, 404–411 https://doi.org/10.2478/BF02475915
  18. S. Mori, On degrees and genera of curves on smooth quartic surfaces in P^3$, Nagoya Math. J. 96 (1984), 127–132
  19. B. G. Moisezon, Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 225–268
  20. P. E. Newstead, A space curve whose normal bundle is stable, J. London Math. Soc. (2) 28 (1983), no. 3, 428–434 https://doi.org/10.1112/jlms/s2-28.3.428
  21. C. Okonek, M. Schneider, and H. Splinder, Vector Bundles on Complex Projective Spaces, Progress in Mathematics, 3. Birkhauser, Boston, Mass., 1980 https://doi.org/10.1090/S0273-0979-1981-14946-6
  22. G. Ottaviani, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl. (4) 155 (1989), 317–341 https://doi.org/10.1007/BF01765948
  23. A. Vogeelar, Constructing vector bundles from codimension-two subvarieties, PhD thesis, Leiden, 1978

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