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Would someone please give an example of a smooth, proper but non-projective curve $C/S$, where $S$ is affine and connected? I believe that whatever your example, $C/S$ must have genus $1$, admit no sections, and $S$ cannot be of dimension $0$ or regular of dimension $1$.


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See Ch. XIII, 3.1 of Raynaud's thesis (SLN 119): uses 1-dimensional local integral noetherian $S$. Or was that stolen from your local math library? – BCnrd Oct 8 '10 at 2:52
Lovely! Thanks, Brian! – Thanos D. Papaïoannou Oct 8 '10 at 2:58
Raynaud also gives an example of a "curve of genus 1", over a tw-dimensional local normal base, which is an algebraic space but not a scheme: see XIII, 3.2 and remark XIII, 3.1b) following it. – Laurent Moret-Bailly Oct 8 '10 at 8:31

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