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On some moduli spaces of stable vector bundles on cubic and quartic threefolds. (English) Zbl 1144.14036

Rank 2 ACM vector bundles [see the Introduction or A. Beauville, Mich. Math. J. 48, Spec. Vol., 39–64 (2000; Zbl 1076.14534)] on smooth quartic threefolds were classified by C. Madonna [Rev. Mat. Complut. 13, No. 2, 287–301 (2000; Zbl 0981.14019)], and the rank 2 ACM vector bundles on the smooth cubic threefolds were classified by E. Arrondo and L. Costa [Commun. Algebra 28, No. 8, 3899–3911 (2000; Zbl 1004.14010)]. Some of the moduli spaces on cubic and quartic threefolds have been studied in more detail by different authors (see the Introduction), and this paper continues the study of these moduli spaces.
The two main results proved here are the following (see Theorems 1-2): The moduli spaces \(M_X(2;1,2)\) and \(M_X(2;0,1)\) on a smooth cubic hypersurface \(X\) are both isomorphic to the Fano surface of lines on \(X\). On the general quartic threefold \(X\): (1) any component of the moduli space \(M_X(2;2,8)\) that contains only ACM bundles is smooth and of dimension 5; (2) \(M_X(2;1,3)\) is isomorphic to the curve of lines on \(X\); (3) \(M_X(2;0,2)\) is isomorphic to the Fano surface of conics on \(X\).

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J45 Fano varieties
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References:

[1] Beauville, A., Determinantal hypersurfaces, Michigan Math. J., 48, 39-64 (2000) · Zbl 1076.14534
[2] Beauville, A., Vector bundles on the cubic threefold, (Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000). Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), Contemp. Math., vol. 312 (2002), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 71-86 · Zbl 1056.14059
[3] J. Biswas, G.V. Ravindra, Arithmetically Cohen-Macaulay vector bundles on a general complete intersection of sufficiently high multi-degree. Preprint; J. Biswas, G.V. Ravindra, Arithmetically Cohen-Macaulay vector bundles on a general complete intersection of sufficiently high multi-degree. Preprint · Zbl 1200.14082
[4] Buchweitz, R.-O.; Greuel, G.-M.; Schreyer, F.-O., Cohen-Macaulay modules on hypersurface singularities. II, Invent. Math., 88, 1, 165-182 (1987) · Zbl 0617.14034
[5] Clemens, C. H.; Griffiths, P. A., The intermediate Jacobian of the cubic threefold, Ann. of Math., 2, 95, 281-356 (1972) · Zbl 0214.48302
[6] Chiantini, Luca; Madonna, Carlo, ACM bundles on a general quintic threefold, Matematiche (Catania), 55, 2, 239-258 (2000), Dedicated to Silvio Greco on the occasion of his 60th birthday (Catania, 2001, 2002) · Zbl 1165.14304
[7] Chiantini, Luca; Madonna, Carlo, A splitting criterion for rank 2 bundles on a general sextic threefold, Int. J. Math., 15, 4, 341-359 (2004) · Zbl 1059.14054
[8] Chiantini, L.; Madonna, C. K., ACM bundles on general hypersurfaces in \(P^5\) of low degree, Collect. Math., 56, 1, 85-96 (2005) · Zbl 1071.14044
[9] Collino, A., Lines on quartic threefolds, J. Lond. Math. Soc., 19, 257-267 (1979) · Zbl 0432.14024
[10] Collino, A.; Murre, J. P.; Welters, G. E., On the family of conics lying on a quartic threefold, Rend. Sem. Mat. Univ. Politec. Torino, 38, 151-181 (1980) · Zbl 0474.14023
[11] Druel, S., Espace des modules des faisceaux de rang 2 semi-stables de classes de Chern \(c_1 = 0, c_2 = 2\) et \(c_3 = 0\) sur la cubique de \(P^4\), Internat. Math. Res. Not., 19, 985-1004 (2000) · Zbl 1024.14004
[12] Eisenbud, David, (Commutative algebra. With a view toward algebraic geometry. Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (1995), Springer-Verlag: Springer-Verlag New York) · Zbl 0819.13001
[13] Harris, J.; Roth, M.; Starr, J., Curves of small degree on cubic threefolds, Rocky Mount. J. Math., 35, 761-817 (2005) · Zbl 1080.14008
[14] Iliev, A.; Markushevich, D., The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14, Doc. Math., 5, 23-47 (2000) · Zbl 0938.14021
[15] Iliev, A.; Markushevich, D., Quartic 3-fold: Pfaffians, vector bundles, and half-canonical curves, Michigan Math. J., 47, 385-394 (2000) · Zbl 1077.14551
[16] Knörrer, H., Cohen-Macaulay modules on hypersurface singularities I, Invent. Math., 88, 153-164 (1987) · Zbl 0617.14033
[17] Madonna, C., A splitting criterion for rank 2 vector bundles on hypersurfaces in \(P^4\), Rend. Semin. Mat. Univ. Politec. Torino, 56, 43-54 (2000) · Zbl 0957.14012
[18] Markushevich, D.; Tikhomirov, A. S., The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geom., 10, 37-62 (2001) · Zbl 0987.14028
[19] Maruyama, M., Boundedness of semistable sheaves of small ranks, Nagoya Math. J., 78, 65-94 (1980) · Zbl 0456.14011
[20] Mohan Kumar, N.; Rao, A. P.; Ravindra, G. V., Arithmetically Cohen-Macaulay bundles on hypersurfaces, Comment. Math. Helv. Comment. Math. Helv., 82, 4, 829-843 (2007) · Zbl 1131.14047
[21] Mohan Kumar, N.; Rao, A. P.; Ravindra, G. V., Arithmetically Cohen-Macaulay bundles on three dimensional hypersurfaces, Int. Math. Res. Not. IMRN, 8, 11 (2007), Art. ID rnm025 · Zbl 1132.14040
[22] N. Mohan Kumar, A.P. Rao, G.V. Ravindra, On codimension 2 ACM subvarieties in hypersurfaces. Preprint; N. Mohan Kumar, A.P. Rao, G.V. Ravindra, On codimension 2 ACM subvarieties in hypersurfaces. Preprint · Zbl 1181.14025
[23] Mukai, S., Symplectic structure of the moduli space of sheaves on an abelian or \(K 3\) surface, Invent. Math., 77, 101-116 (1984) · Zbl 0565.14002
[24] C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, in: Progress in Mathematics, vol. 3, Birkhäuser; C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, in: Progress in Mathematics, vol. 3, Birkhäuser · Zbl 0438.32016
[25] G.V. Ravindra, Curves on threefolds and a conjecture of Griffiths-Harris. Preprint; G.V. Ravindra, Curves on threefolds and a conjecture of Griffiths-Harris. Preprint
[26] Welters, G. E., Abel-Jacobi isogenies for certain types of Fano threefolds, (Mathematical Centre Tracts, vol. 141 (1981), Mathematisch Centrum: Mathematisch Centrum Amsterdam) · Zbl 0474.14028
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