Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
38(1), 95 - 98 (1997)

On the Generalized Theta Divisor

Georg Hein

Institut für Reine Mathematik, Humboldt-Universität Berlin Ziegelstr. 13A, D-10099 Berlin

Abstract: Let $X$ be a smooth curve of genus $g$ defined over the complex numbers. We denote by $SU_X(n)$ the moduli space of semistable $X$-vector bundles of rank $n$. $SU_X(n)$ is a projective variety and its Picard group is cyclic. [J.M. Drezet, M.S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. math. 97 (1989), p. 53-94, Zbl689.14012] Let $L$ be its ample generator. Since $L$ is ample a high tensor power $L^k$ of $L$ is globally generated. For $n=2$ it was shown in [M. Raynaud, Section des fibrés vectoriels sur une courbe, Bull. soc. math. France 110(1982), p. 103-125, Zbl505.14011] that $L$ itself is globally generated. We give a new proof of this result. However, this result does not extend to the case $n>2$. For $n>2$, we show that for any $k>(g-1)(n-1)^2$ the the linebundle $L^k$ is globally generated. In order to do so we follows the idea of [J. Le Potier, Module des fibrés semistables et fonctions thêta, preprint (1993)] by showing that a rank $k$-bundle $E$ of degree $n(g-1)$ defines a section in $L^k$. The corresponding divisor $D_E$ is (set theoretically) the set of all bundles $F \in SU_X(n)$ which satisfy $H^*(X, E \otimes F) \not= 0$. Hence we have to show that for any $F \in SU_X(n)$ there exists such a vector bundle $E$.

Keywords: generalized Theta divisor, vector bundles, base point freeness, moduli of vector bundles

Classification (MSC91): 14H60, 14H10

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