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Is there an example of a flat proper morphism of schemes $X\rightarrow S$ whose fibers are geometrically connected, reduced and have dimension 1, but which is not itself finitely presented? What happens if we additionally require that the base scheme $S$ is connected and the fibers are stable curves of a constant (arithmetic) genus?

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    $\begingroup$ Google translate: Is there an example of a flat proper morphism of the schemes $X \to S$, whose layers are geometrically connected, reduced and have dimension 1, but which is not of course representable? What if we additionally require that the base be connected, and the layers be stable curves of a constant (arithmetic) kind? $\endgroup$ – YCor Jun 3 at 6:37
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    $\begingroup$ Would someone knowledgable please edit the translation into the question? Usually we do not encourage questions in russian. $\endgroup$ – András Bátkai Jun 3 at 6:42
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    $\begingroup$ @YCor I think it is somewhat amusing that in Russian the word for "of course" is the same as the word for "finitely". $\endgroup$ – user141414 Jun 3 at 8:56
  • $\begingroup$ These words are pronounced differently though: [конешно] vs [конечно]. $\endgroup$ – Daniil Rudenko Jun 4 at 5:57
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Regarding the first question, this may be useful. It is claimed that there exists "a proper flat morphism of schemes $X \rightarrow S$ each of whose fibres is isomorphic to either $\mathbb{P}^1_s$ or to the vanishing locus of $X_1 X_2$ in $\mathbb{P}^2_s$ which is not of finite presentation."

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