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Let $X$ be a projective Noetherian scheme over $\mathbb{C}$. Is it true that $H^0(\mathcal{O}_X(-t))=0$ for any $t>0$?

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Note that $O_X(1)$ isn't well-defined unless you specify an embedding in projective space. But yes, the dual of a very ample line bundle (or even just a nontrivial effective line bundle) cannot have nonzero global sections: if $L$ and $L^*$ both had nonconstant global sections, we could multiply them to get a nonconstant section of $O_X$.

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@Dubh: Thank you very much for the answer. Could you please elaborate on the reason. You can assume an embeeding into a projective space if necessary. – user46578 Feb 16 '14 at 13:42

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