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Imagine a projective, noetherian and flat family of curves $C \rightarrow S$ (i.e. every geometric fiber is a curve, this is a integral, non singular, proper scheme of dimension 1 over an algebraically closed field) which is smooth. By the stein-factorization there is a scheme $S'$, such that $C\rightarrow S$ decomposes to $C\rightarrow S' \rightarrow S$ where $C \rightarrow S'$ is a projective morphism with connected fibers and $S' \rightarrow S$ is a morphism of finite type.

The paper I'm reading now states, that $S'$ and $S$ are isomorphic on an open subset.

  1. Is this the case?

  2. Why is this the case, especially, which of the many nice properties from above gives us this isomorphism.

Thanks

Frank

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