Groupe de Picard des variétés de modules de faisceaux semi-stable sur \({\mathbb{P}}_ 2({\mathbb{C}})\). (Picard group of the moduli varieties of semi- stable sheaves on \({\mathbb{P}}_ 2({\mathbb{C}}))\). (French) Zbl 0616.14006

The subject of this paper is the Picard group of the moduli variety \(M(r,c_ 1,c_ 2)\) of semi-stable algebraic sheaves on \({\mathbb{P}}_ 2({\mathbb{C}})\), of \(rank\quad r\) and Chern classes \(c_ 1, c_ 2\). The first result is that if \(M(r,c_ 1,c_ 2)\) is locally factorial, so \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to the group of linear equivalence classes of Weil divisors of \(M(r,c_ 1,c_ 2)\). There is a unique map \(\delta: {\mathbb{Q}}\to {\mathbb{Q}}\) such that \(\dim (M(r,c_ 1,c_ 2))>0\) if and only if \((c_ 2-(r-1)c^ 2_ 1/2r)/r\geq \delta (c_ 1/r)\). Then if one has equality, \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to \({\mathbb{Z}}\), and if the inequality is strict, \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to \({\mathbb{Z}}^ 2\). A description of \(Pic(M(r,c_ 1,c_ 2))\) is given, using a subgroup of the Grothendieck group \(K({\mathbb{P}}_ 2)\) of \({\mathbb{P}}_ 2\). It is then possible to compute the canonical bundle \(\omega _ M\) of \(M(r,c_ 1,c_ 2)\).


14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R20 Characteristic classes and numbers in differential topology
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[1] W. BARTH, Moduli of vector bundles on the projective plane, Invent. Math., 42 (1977), 63-91. · Zbl 0386.14005
[2] A.A. BEILINSON, Coherent sheaves on pn and problems of linear algebra, Funkt. Anal. i Ego Pril., vol. 12, No 3 (1978), 68-69. · Zbl 0402.14006
[3] A. BOREL, J.P. SERRE, Le théorème de Riemann-Roch, Bull. Soc. Math. de France, 86 (1858), 97-136. · Zbl 0091.33004
[4] E. BRIESKORN, Uber holomorphe pn-Bündel über P1, Math, Ann., 157 (1967), 343-357. · Zbl 0128.17003
[5] J. BRUN, A. HIRSCHOWITZ, Droites de saut des fibrés de rang élevé sur P2, Math, Zeits., 181 (1982), 171-178. · Zbl 0473.14005
[6] J.M. DREZET, Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur P2 (C), Preprint, 1985.
[7] J.M. DREZET, J. LE POTIER, Fibrés stables et fibrés exceptionnels sur P2, Ann. Scient. Ec. Norm. Sup., 18 (1985), 193-244. · Zbl 0586.14007
[8] G. ELLINGSRUD, Sur l’irréductibilité du module des fibrés stables sur P2, Math. Zeits., 182 (1983), 189-192. · Zbl 0526.14003
[9] G. ELLINGSRUD, S.A. STRØMME, The Picard group of the moduli for stable rank 2 vector bundles on P2 with odd Chern class, Preprint, Oslo, 1979.
[10] W. FULTON, Intersection theory, Springer Verlag, 1984. · Zbl 0541.14005
[11] D. GIESEKER, On the moduli of vector bundles on an algebraic surface, Ann. of Math., 106 (1977), 45-60. · Zbl 0381.14003
[12] A. GROTHENDIECK, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. of Math., 79 (1957), 121-138. · Zbl 0079.17001
[13] R. HARTSHORNE, Algebraic geometry, Grad. Texts in Math., 52, Springer-Verlag, 1977. · Zbl 0367.14001
[14] A. HIRSCHOWITZ, Rank techniques and jump stratifications. Actes du congrès “Vector Bundles on Algebraic Varieties” Bombay, 1984. · Zbl 0682.14009
[15] G. HORROCKS, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc., 14 (1964), 689-713. · Zbl 0126.16801
[16] K. HULEK, On the classification of stable rank-r vector bundles over the projective plane. In : Vector bundles and differential equations (A. Hirschowitz ed.). Proceedings (Nice 1979), Progress in Math., 7, Birkhäuser, 1980. · Zbl 0446.14006
[17] J. LE POTIER, Sur le groupe de Picard de l’espace de modules de fibrés stables sur P2, Ann. Scient. Ec. Norm. Sup., 14 (1981), 141-155. · Zbl 0482.14006
[18] J. LE POTIER, Fibrés stables de rang 2 sur P2(C), Math. Ann., 241 (1979), 217-256. · Zbl 0405.14008
[19] M. MARUYAMA, Moduli of stable sheaves II, J. Math. Kyoto Univ., 18 (1978), 557-614. · Zbl 0395.14006
[20] H. MATSUMURA, Commutative algebra, W.A. Benjamin Co., New York, 1980. · Zbl 0441.13001
[21] D. MUMFORD, J. FOGARTY, Geometric invariant theory, Erg. der Math. und ihre Grenzg., 34, Springer-Verlag, 1982. · Zbl 0504.14008
[22] P.E. NEWSTEAD, Introduction to moduli problems and orbit spaces, Tata Inst., Lect. Notes, 51, Springer-Verlag, 1978. · Zbl 0411.14003
[23] P.E. NEWSTEAD, Rationality of moduli spaces of vector bundles, Math. Ann., 215 (1975), 251-268. · Zbl 0288.14003
[24] C.S. SESHADRI, Mumford’s conjecture for GL(2) and applications, Alg. Geom., Bombay Coll., (1968-1969), 347-371. · Zbl 0194.51702
[25] S.A. STRØMME, Ample divisors on fine moduli spaces on the projective plane, Math. Zeits., 187 (1984), 405-423. · Zbl 0533.14006
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