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It is interesting to study the moduli space $M_g$ of smooth projective curves of genus $g$.

Why not smooth affine curves of genus $g$?

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What is the genus of an affine curve? – Zhen Lin Feb 6 '12 at 8:55
It has a function field. Then you take the unique smooth projective curve with this function field. – seporhau Feb 6 '12 at 9:04
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Ow I see where the difficulty is. Any non-empty open subscheme of a smooth affine curve is a smooth affine curve....of the same genus. – seporhau Feb 6 '12 at 9:06

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