Poincaré bundles for projective surfaces

Mestrano, Nicole

Annales de l'Institut Fourier, Tome 35 (1985), p. 217-249 / Harvested from Numdam

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Abstract

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 $2 c_{1}^{'}$ et si $c_{2} - \frac{1}{4} c_{1}^{2}$ est pair, au moins égal à $H^{2} + H K + 4$ , il n’y a pas de fibré de Poincaré sur $M_{H} (c_{1}, c_{2}) \times X$ . Par contre s’il existe $c_{1}^{'}$ tel que le nombre $c_{1} \cdot c_{1}^{'}$ soit impair, ou bien si $\frac{1}{2} c_{1}^{2} - \frac{1}{2} c_{1} \cdot K - c_{2}$ est impair, alors il y a un fibré de Poincaré sur $M_{H} (c_{1}, c_{2}) \times X$ .

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 $2 c_{1}^{'}$ and if $c_{2} - \frac{1}{4} c_{1}^{2}$ is even, greater or equal to $H^{2} + H K + 4$ , then there is no Poincaré bundle on $M_{H} (c_{1}, c_{2}) \times X$ . Conversely, if there exists $c_{1}^{'}$ such that the number $c_{1}^{'} \cdot c_{1}$ is odd or if $\frac{1}{2} c_{1}^{2} - \frac{1}{2} c_{1} \cdot K - c_{2}$ is odd, then there exists a Poincaré bundle on $M_{H} (c_{1}, c_{2}) \times X$ .

Publié le : 1985-01-01

MR 87c:14019 | Zbl 0532.14005

DOI : https://doi.org/10.5802/aif.1015

@article{AIF_1985__35_2_217_0,
     author = {Mestrano, Nicole},
     title = {Poincar\'e bundles for projective surfaces},
     journal = {Annales de l'Institut Fourier},
     volume = {35},
     year = {1985},
     pages = {217-249},
     doi = {10.5802/aif.1015},
     mrnumber = {87c:14019},
     zbl = {0532.14005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1985__35_2_217_0}
}

Mestrano, Nicole. Poincaré bundles for projective surfaces. Annales de l'Institut Fourier, Tome 35 (1985) pp. 217-249. doi : 10.5802/aif.1015. http://gdmltest.u-ga.fr/item/AIF_1985__35_2_217_0/

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