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

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.

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.

@article{AIF_1985__35_2_217_0,
     author = {Mestrano, Nicole},
     title = {Poincar\'e bundles for projective surfaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {35},
     number = {2},
     year = {1985},
     pages = {217-249},
     doi = {10.5802/aif.1015},
     zbl = {0532.14005},
     mrnumber = {87c:14019},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1985__35_2_217_0}
}
Mestrano, Nicole. Poincaré bundles for projective surfaces. Annales de l'Institut Fourier, Volume 35 (1985) no. 2, pp. 217-249. doi : 10.5802/aif.1015. http://www.numdam.org/item/AIF_1985__35_2_217_0/

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