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!