Poincaré bundles for projective surfaces

Mestrano, Nicole

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 $c_1^{\prime }$ tel que $c_1$ est numériquement équivalent à $2c_1^{\prime }$ et si $c_2-\frac{1}{4}{c}_1^2$ est pair, au moins égal à $H^2+HK+4$, il n’y a pas de fibré de Poincaré sur $M_H(c_1,c_2)\times X$. Par contre s’il existe $c_1^{\prime }$ tel que le nombre $c_1·c_1^{\prime }$ soit impair, ou bien si $\frac{1}{2}{c}_1^2-\frac{1}{2}{c}_1·K-c_2$ 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 $c_1^{\prime }$ such that $c_1$ is numerically equivalent to $2c_1^{\prime }$ and if $c_2-\frac{1}{4}{c}_1^2$ is even, greater or equal to $H^2+HK+4$, then there is no Poincaré bundle on $M_H(c_1,c_2)\times X$. Conversely, if there exists $c_1^{\prime }$ such that the number $c_1^{\prime }·c_1$ is odd or if $\frac{1}{2}{c}_1^2-\frac{1}{2}{c}_1·K-c_2$ is odd, then there exists a Poincaré bundle on $M_H(c_1,c_2)\times X$.

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