### Stability of Arakelov bundles and tensor products without global sections.

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Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M r,Lss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M r,Lss) = ℤ, identify the ample generator, and deduce that M r,Lss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant...

Let $X$ and ${X}^{\prime}$ be compact Riemann surfaces of genus $\ge 3$, and let $G$ and ${G}^{\prime}$ be nonabelian reductive complex groups. If one component ${\mathcal{M}}_{G}^{d}\left(X\right)$ of the coarse moduli space for semistable principal $G$–bundles over $X$ is isomorphic to another component ${\mathcal{M}}_{{G}^{\prime}}^{{d}^{\prime}}\left({X}^{\prime}\right)$, then $X$ is isomorphic to ${X}^{\prime}$.

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