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One may define a curve (e.g. separated scheme of finite type of dim. 1) over an algebraically closed field, as done in Hartshorne's book. A weaker assumption, which is used commonly, is to define a curve $C$ over a perfect field $k$.

What properties does such a curve have compared to a curve over an arbitrary field, and compared to a curve over an algebraically closed field, maybe in terms of singularities?

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    $\begingroup$ Over a perfect field $k$, any reduced scheme $X$ of finite type is $k$-smooth on a dense open subset since for each (reduced) irreducible component $X_i$ of $X$ the function field $k(X_i)$ admits a separating transcendence basis (as for any finitely generated field over a perfect field). This fails over every imperfect field $k$ (e.g., $y^2=x^p-a$ for $a\in k-k^p$ with ${\rm{char}}(k)=p>0$). More conceptually, "reduced" is the same as "geometrically reduced over $k$" for such $X$, since $\overline{k}/k$ is a directed union of finite etale extensions of $k$ (by definition of perfectness). $\endgroup$
    – nfdc23
    Jun 13, 2016 at 10:49
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    $\begingroup$ The example of nfdc23 also shows that, over an imperfect field $k$, a finite type $k$-scheme that is regular need not be smooth (even if it is geometrically reduced). $\endgroup$
    – Jason Starr
    Jun 13, 2016 at 11:24

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