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Let $Y$ be a proper variety over a field $k$. Suppose that $\mathcal{L}$ is an ample line bundle on Y and suppose that $\mathcal{L}$ is isomorphic to the trivial bundle. What can I conclude about the dimension of the variety?

From the assumption I can (hopefully) deduce that the trivial bundle $\mathcal{O}_Y$ is not only ample: it is indeed very ample (relative to Spec$(k)$), so that there is an immersion $i\colon Y \to \mathbb{P}^n$, for some $r$ such that $\mathcal{O}_Y \cong i^{*}(\mathcal{O}(1))$. My feeling is that this should imply that the variety is affine - that forces the dimension of $Y$ to be $0$, being proper - but I don't see a proof.

Thank you!

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up vote 3 down vote accepted

The assumption that the structure sheaf is ample means that $Y$ is quasi-affine (see for example Proposition 5.1.2 in EGA II), which is equivalent to the canonical morphism $i:Y\to\mathop{\mathrm{Spec}}(\mathcal{O}_Y(Y))$ being an open immersion. Now $Y$ being proper implies that $i$ is proper and hence it is a closed immersion. Therefore $Y$ is indeed affine.

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Thank you for the answer. I actually missed the cited result in EGA II. –  FedeB Mar 3 '12 at 15:19
    
....................nice. –  S123 Mar 3 '12 at 16:11
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