no. 2
Poincaré bundles for projective surfaces
Annales de l'Institut Fourier, Tome 35 (1985) no. 2, pp. 217-249.

Soit X une surface projective lisse, K le diviseur canonique, H un diviseur très ample et M H (c 1 ,c 2 ) l’espace des modules de fibrés vectoriels de rang deux H-stables, de classes de Chern c 1 et c 2 . On démontre que s’il existe c 1 ′ tel que c 1 est numériquement équivalent à 2c 1 ′ et si c 2 -1 4c 1 2 est pair, au moins égal à H 2 +HK+4, il n’y a pas de fibré de Poincaré sur M H (c 1 ,c 2 )×X. Par contre s’il existe c 1 ′ tel que le nombre c 1 ·c 1 ′ soit impair, ou bien si 1 2c 1 2 -1 2c 1 ·K-c 2 est impair, alors il y a un fibré de Poincaré sur M H (c 1 ,c 2 )×X.

Let X be a smooth projective surface, K the canonical divisor, H a very ample divisor and M H (c 1 ,c 2 ) the moduli space of rank-two vector bundles, H-stable with Chern classes c 1 and c 2 . We prove that, if there exists c 1 ′ such that c 1 is numerically equivalent to 2c 1 ′ and if c 2 -1 4c 1 2 is even, greater or equal to H 2 +HK+4, then there is no Poincaré bundle on M H (c 1 ,c 2 )×X. Conversely, if there exists c 1 ′ such that the number c 1 ′ ·c 1 is odd or if 1 2c 1 2 -1 2c 1 ·K-c 2 is odd, then there exists a Poincaré bundle on M H (c 1 ,c 2 )×X.

@article{AIF_1985__35_2_217_0,
     author = {Mestrano, Nicole},
     title = {Poincar\'e bundles for projective surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {217--249},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {35},
     number = {2},
     year = {1985},
     doi = {10.5802/aif.1015},
     zbl = {0532.14005},
     mrnumber = {87c:14019},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1015/}
}
TY  - JOUR
AU  - Mestrano, Nicole
TI  - Poincaré bundles for projective surfaces
JO  - Annales de l'Institut Fourier
PY  - 1985
SP  - 217
EP  - 249
VL  - 35
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1015/
DO  - 10.5802/aif.1015
LA  - en
ID  - AIF_1985__35_2_217_0
ER  - 
%0 Journal Article
%A Mestrano, Nicole
%T Poincaré bundles for projective surfaces
%J Annales de l'Institut Fourier
%D 1985
%P 217-249
%V 35
%N 2
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1015/
%R 10.5802/aif.1015
%G en
%F AIF_1985__35_2_217_0
Mestrano, Nicole. Poincaré bundles for projective surfaces. Annales de l'Institut Fourier, Tome 35 (1985) no. 2, pp. 217-249. doi : 10.5802/aif.1015. https://aif.centre-mersenne.org/articles/10.5802/aif.1015/

[1] W. Barth, Some properties of stable rank-2 vector bundles on Pn, Math. Ann., 226 (1977). | MR | Zbl

[2] G. Ellingsrud, S. A. Strømme, On the moduli space for stable rank-2 vector bundles on P2 (preprint Oslo). | Zbl

[3] R. Harsthørne, Algebraic geometry, Springer, 1977. | Zbl

[4] R. Hartshorne, Stable reflexive sheaves, Math. Ann., 254 (1980), 121-176. | MR | Zbl

[5] A. Hirschowitz, M. S. Narasimhan, Fibrés de 't Hooft spéciaux et applications, Enumerative geometry and classical algebraic geometry, Progress in Math. (1982). | Zbl

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

[7] M. Maruyama, On a family of algebraic vector bundles, Number theory, Alg. Geo. and Com. Alg., Tokyo (1973). | MR | Zbl

[8] M. Maruyama, Moduli of stable sheaves II, J. of Math. of Kyoto Univ., Vol. 18 (1978). | MR | Zbl

[9] M. Maruyama, Elementary transformation in the theory of vector bundles, Lecture Notes, 961, Alg. Geo., Springer Verlag, 1981. | Zbl

[10] N. Mestrano, Sections rationnelles de morphismes algébriques (preprint, Nice), 1983.

[11] N. Mestrano, S. Ramanan, Poincaré bundles for families of curves (preprint, 1984). | Zbl

[12] M. S. Narasimhan, S. Ramanan, Vector bundles on curves in algebraic geometry, Bombay Colloquium, 1968. | Zbl

[13] P. E. Newstead, A non existence theorem for families of stable bundles, Jour. Lond. Math. Soc. | Zbl

[14] S. Ramanan, The moduli space of vector bundles over an algebraic curve, Math. Ann., 200 (1973). | MR | Zbl

[15] R. L. E. Schwarzenberger, Vector bundles on algebraic surfaces, Proc. London Math. Soc., Vol. 11 (1961). | MR | Zbl

[16] F. Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. Jour., 47 (1972). | MR | Zbl

Cité par Sources :