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Let $X$ and $Y$ be two proper smooth connected curves over $S = \text{Spec}\ k$, where $k$ is an algebraically closed field.

Let $f$ be an $S$-morphism $X \to Y$, then in [KM, p74] it is stated that $f$ is either finite flat or constant. I do not see why/how. Also a search did not give me results on where to find a proof.

When assuming that $X$ and $Y$ are elliptic curves, I do see:

  • $X$ and $Y$ are projective over $S$
  • Therefore $f$ is projective
  • And the statement (intuitively) makes sense to me over $\mathbb{C}$.

But I do not see why this is true in the more general setting.

[KM] : N. M. Katz — B. Mazur, Aritmetic Moduli of Elliptic Curves. Annals of Mathematics Studies, Princeton University Press, 1985

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I think it is proved in Hartshorne II.6.

But it is not difficult (fill in the details!): If $f$ is not constant, it is surjective. Then, since $Y$ is a Dedekind scheme, $f$ is flat. Since it is quasi-finite and proper, it is finite.

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  • $\begingroup$ In Hartshorne II.6.8 it indeed proves the finiteness. Thanks! How about the flatness? $\endgroup$
    – jmc
    Commented Mar 14, 2012 at 19:23
  • $\begingroup$ See e.g. Görtz, Wedhorn, Proposition 14.14. $\endgroup$
    – user19475
    Commented Mar 14, 2012 at 19:33

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