Biblioteca de la Universidad Complutense de Madrid

Curves and vector bundles on quartic threefolds


Arrondo Esteban, Enrique y Maddona, Carlo G. (2009) Curves and vector bundles on quartic threefolds. Journal of the Korean Mathematical Society, 46 (3). pp. 589-608. ISSN 0304-9914

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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.

Tipo de documento:Artículo
Palabras clave:Intermediate cohomology; Criterion; Quartic threefold; ACM bundle; Projectively normal curve
Materias:Ciencias > Matemáticas > Geometria algebraica
Código ID:14767

[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,


[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-


[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,


[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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