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Let $X$ be a smooth projective curve of genus $g>2$ over an algebraically closed field $k$. Does there exist a line bundle $L$ on $X$ of degree $(g-1)$ such that $H^0(X,L)=0$?

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The locus of $L$ with $H^0(X,L) \ne 0$ is the image of the natural map $X^{g-1} \to Pic^{g-1}(X)$. Its image is at most $(g-1)$-dimensional, hence its complement is nonempty.

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