CURVES AND VECTOR BUNDLES ON QUARTIC THREEFOLDS

Title & Authors
CURVES AND VECTOR BUNDLES ON QUARTIC THREEFOLDS

Abstract
In this paper we study arithmetically Cohen-Macaulay (ACM for short) vector bundles $\small{\varepsilon}$ of rank k $\small{\geq}$ 3 on hypersurfaces $\small{X_r\;{\subset}\;{\mathbb{P}}^4}$ of degree r $\small{\geq}$ 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 $\small{\varepsilon}$ we derive a list of possible Chern classes ($\small{c_1}$, $\small{c_2}$, $\small{c_3}$) which may arise in the cases of rank k = 3 and k = 4, when r = 4 and we give several examples.
Keywords
quartic threefold;ACM bundle;projectively normal curve;
Language
English
Cited by
1.
STABLE ULRICH BUNDLES, International Journal of Mathematics, 2012, 23, 08, 1250083
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

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 $V_{22}$, 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

11.
G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964), 689–713

12.
A. Iliev and D. Markushevich, Quartic 3-fold: Pfaffians, vector bundles, and halfcanonical curves, Michigan Math. J. 47 (2000), no. 2, 385–394

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$, Rev. Mat. Complut. 13 (2000), no. 2, 287–301 16. C. Madonna, ACM vector bundles on prime Fano threefolds and complete intersection Calabi-Yau threefolds, Rev. Roumaine Math. Pures Appl. 47 (2002), no. 2, 211–222 17. C. Madonna, Rank 4 vector bundles on the quintic threefold, Cent. Eur. J. Math. 3 (2005), no. 3, 404–411 18. S. Mori, On degrees and genera of curves on smooth quartic surfaces in P^3$, 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

21.
C. Okonek, M. Schneider, and H. Splinder, Vector Bundles on Complex Projective Spaces, Progress in Mathematics, 3. Birkhauser, Boston, Mass., 1980

22.
G. Ottaviani, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl. (4) 155 (1989), 317–341

23.
A. Vogeelar, Constructing vector bundles from codimension-two subvarieties, PhD thesis, Leiden, 1978