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.




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