Groupe de Picard des variétés de modules de faisceaux semi-stables sur ${\mathbb {P}}_2({\mathbb {C}})$

Drezet, Jean-Marc

doi:10.5802/aif.1143

Groupe de Picard des variétés de modules de faisceaux semi-stables sur

ℙ_{2} (ℂ)

Drezet, Jean-Marc

Annales de l'Institut Fourier, Tome 38 (1988) no. 3, pp. 105-168.

Résumé
Abstract

Le sujet de cet article est le groupe de Picard de la variété de modules $M (r, c_{1}, c_{2})$ des faisceaux algébriques semi-stables de rang $r$ et de classes de Chern $c_{1}, c_{2}$ sur $P_{2} (C)$ . Le premier résultat est que $M (r, c_{1}, c_{2})$ est localement factorielle, ce qui permet d’identifier Pic $(M (r, c_{1}, c_{2}))$ et le groupe des classes d’équivalence linéaire des diviseurs de Weil de $M (r, c_{1}, c_{2}))$ . Il existe une unique application $δ : Q \to Q$ telle que dim $(M (r, c_{1}, c_{2})) > 0$ si et seulement si $(c_{2} - (r - 1) c_{1}^{2} / 2 r) / r > δ (c_{1} / r)$ . Si on a égalité, Pic $(M (r, c_{1}, c_{2}))$ est isomorphe à $Z$ , et si l’inégalité est stricte, Pic $(M (r, c_{1}, c_{2}))$ est isomorphe à $Z^{2}$ . On donne ensuite une description de Pic $(M (r, c_{1}, c_{2}))$ faisant intervenir un sous-groupe du groupe de Grothendieck $K (P_{2})$ de $P_{2}$ . On peut enfin calculer le fibré canonique $ω_{M}$ de $M (c_{1}, c_{2})$ .

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 $P_{2} (C)$ , of $r a n k r$ 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 $δ : Q \to Q$ such that $dim (M (r, c_{1}, c_{2})) > 0$ if and only if $(c_{2} - (r - 1) c_{1}^{2} / 2 r) / r \geq δ (c_{1} / r)$ . Then if one has equality, $Pic (M (r, c_{1}, c_{2}))$ is isomorphic to $Z$ , and if the inequality is strict, $P i c (M (r, c_{1}, c_{2}))$ is isomorphic to $Z^{2}$ . A description of $Pic (M (r, c_{1}, c_{2}))$ is given, using a subgroup of the Grothendieck group $K (P_{2})$ of $P_{2}$ . It is then possible to compute the canonical bundle $ω_{M}$ of $M (r, c_{1}, c_{2})$ .

MR Zbl | 4 citations dans Numdam

DOI : 10.5802/aif.1143

@article{AIF_1988__38_3_105_0,
     author = {Drezet, Jean-Marc},
     title = {Groupe de {Picard} des vari\'et\'es de modules de faisceaux semi-stables sur ${\mathbb {P}}_2({\mathbb {C}})$},
     journal = {Annales de l'Institut Fourier},
     pages = {105--168},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {3},
     year = {1988},
     doi = {10.5802/aif.1143},
     mrnumber = {89m:14007},
     zbl = {0616.14006},
     language = {fr},
     url = {https://www.numdam.org/articles/10.5802/aif.1143/}
}

TY  - JOUR
AU  - Drezet, Jean-Marc
TI  - Groupe de Picard des variétés de modules de faisceaux semi-stables sur ${\mathbb {P}}_2({\mathbb {C}})$
JO  - Annales de l'Institut Fourier
PY  - 1988
SP  - 105
EP  - 168
VL  - 38
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - https://www.numdam.org/articles/10.5802/aif.1143/
DO  - 10.5802/aif.1143
LA  - fr
ID  - AIF_1988__38_3_105_0
ER  -

%0 Journal Article
%A Drezet, Jean-Marc
%T Groupe de Picard des variétés de modules de faisceaux semi-stables sur ${\mathbb {P}}_2({\mathbb {C}})$
%J Annales de l'Institut Fourier
%D 1988
%P 105-168
%V 38
%N 3
%I Institut Fourier
%C Grenoble
%U https://www.numdam.org/articles/10.5802/aif.1143/
%R 10.5802/aif.1143
%G fr
%F AIF_1988__38_3_105_0

Drezet, Jean-Marc. Groupe de Picard des variétés de modules de faisceaux semi-stables sur ${\mathbb {P}}_2({\mathbb {C}})$. Annales de l'Institut Fourier, Tome 38 (1988) no. 3, pp. 105-168. doi : 10.5802/aif.1143. https://www.numdam.org/articles/10.5802/aif.1143/

Bibliographie
Cité par

[1] W. Barth, Moduli of vector bundles on the projective plane, Invent. Math., 42 (1977), 63-91. | MR | Zbl

[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

[3] A. Borel, J.P. Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. de France, 86 (1858), 97-136. | Numdam | Zbl

[4] E. Brieskorn, Uber holomorphe Pn-Bündel über P1, Math, Ann., 157 (1967), 343-357. | MR | Zbl

[5] J. Brun, A. Hirschowitz, Droites de saut des fibrés de rang élevé sur P2, Math, Zeits., 181 (1982), 171-178. | MR | Zbl

[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. | Numdam | MR | Zbl

[8] G. Ellingsrud, Sur l'irréductibilité du module des fibrés stables sur P2, Math. Zeits., 182 (1983), 189-192. | MR | Zbl

[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. | MR | Zbl

[11] D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math., 106 (1977), 45-60. | MR | Zbl

[12] A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. of Math., 79 (1957), 121-138. | MR | Zbl

[13] R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52, Springer-Verlag, 1977. | MR | Zbl

[14] A. Hirschowitz, Rank techniques and jump stratifications. Actes du congrès “Vector Bundles on Algebraic Varieties” Bombay, 1984. | Zbl

[15] G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc., 14 (1964), 689-713. | MR | Zbl

[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

[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. | Numdam | MR | Zbl

[18] J. Le Potier, Fibrés stables de rang 2 sur P2(C), Math. Ann., 241 (1979), 217-256. | MR | Zbl

[19] M. Maruyama, Moduli of stable sheaves II, J. Math. Kyoto Univ., 18 (1978), 557-614. | MR | Zbl

[20] H. Matsumura, Commutative algebra, W.A. Benjamin Co., New York, 1980. | MR | Zbl

[21] D. Mumford, J. Fogarty, Geometric Invariant Theory, Erg. der Math. und ihre Grenzg., 34, Springer-Verlag, 1982. | MR | Zbl

[22] P.E. Newstead, Introduction to moduli problems and orbit spaces, Tata Inst., Lect. Notes, 51, Springer-Verlag, 1978. | MR | Zbl

[23] P.E. Newstead, Rationality of moduli spaces of vector bundles, Math. Ann., 215 (1975), 251-268. | MR | Zbl

[24] C.S. Seshadri, Mumford's conjecture for GL(2) and applications, Alg. Geom., Bombay Coll., (1968-1969), 347-371. | MR | Zbl

[25] S.A. Strømme, Ample divisors on fine moduli spaces on the projective plane, Math. Zeits., 187 (1984), 405-423. | MR | Zbl

Pedchenko, Dmitrii The Picard Group of the Moduli Space of Sheaves on a Quadric Surface, International Mathematics Research Notices, Volume 2022 (2022) no. 23, p. 18676 | DOI:10.1093/imrn/rnab175
Yuan, Yao Strange Duality on Rational Surfaces II: Higher-Rank Cases, International Mathematics Research Notices, Volume 2020 (2020) no. 10, p. 3153 | DOI:10.1093/imrn/rny133
Yuan, Yao Moduli spaces of 1-dimensional semi-stable sheaves and strange duality on P2, Advances in Mathematics, Volume 318 (2017), p. 130 | DOI:10.1016/j.aim.2017.07.014
Huizenga, Jack Birational geometry of moduli spaces of sheaves and Bridgeland stability, Surveys on Recent Developments in Algebraic Geometry, Volume 95 (2017), p. 101 | DOI:10.1090/pspum/095/01639
Bertram, Aaron; Martinez, Cristian; Wang, Jie The birational geometry of moduli spaces of sheaves on the projective plane, Geometriae Dedicata, Volume 173 (2014) no. 1, p. 37 | DOI:10.1007/s10711-013-9927-1
Perego, Arvid; Rapagnetta, Antonio Factoriality Properties of Moduli Spaces of Sheaves on Abelian and K3 Surfaces, International Mathematics Research Notices, Volume 2014 (2014) no. 3, p. 643 | DOI:10.1093/imrn/rns233
Kaledin, D.; Lehn, M.; Sorger, Ch. Singular symplectic moduli spaces, Inventiones mathematicae, Volume 164 (2006) no. 3, p. 591 | DOI:10.1007/s00222-005-0484-6
Okonek, Ch.; Van de Ven, A. Stable Bundles, Instantons and C∞-Structures on Algebraic Surfaces, Complex Manifolds (1998), p. 197 | DOI:10.1007/978-3-642-61299-2_4
Nakashima, Tohru On the moduli of stable vector bundles on a Hirzebruch surface, Mathematische Zeitschrift, Volume 212 (1993) no. 1, p. 211 | DOI:10.1007/bf02571654
Drezet, Jean-Marc Vari�t�s de modules extr�males de faisceaux semi-stables sur ?2(?), Mathematische Annalen, Volume 290 (1991) no. 1, p. 727 | DOI:10.1007/bf01459270

Cité par 10 documents. Sources : Crossref