I´ve been reading the Deligne-Mumford construction of the moduli of curves with a given genus and I have some questions about the article http://www.numdam.org/article/PMIHES_1969__36__75_0.pdf

1) When the authors talk about a scheme what are they referring to? In the EGA, a scheme is what we call separated scheme, so I don´t know if they are working on the category of separated schemes or just schemes (in our terminology).

2) My second question is about Definition 1.1. They say that a stable curve is a proper flat morphism of schemes $f:X\rightarrow S$ whose geometric fibers are reduced, connected, 1-dimensional schemes such that:

  • $X_{s}$ has only ordinary double points,

  • If $E$ is a non-singular rational component of $X_{s}$ then $E$ meets the other components of $X_{s}$ in more than 2 points;

  • $\rm{dim}\rm{H}^{1}(\mathcal{O}_{X_{s}})=g$

In general, a relative curve is defined as a flat finitely presented morphism of schemes $X\rightarrow S$ of relative dimension 1. My question is if proper+flat in this particular case implies finitely presented. It is the same true if $f:X\rightarrow S$ is a proper and flat morphism whose geometric fibers are complete integral algebraic curves of arithmetic genus $g$?

Thank you for your time.

  • 2
    $\begingroup$ I think de Jong's blog post math.columbia.edu/~dejong/wordpress/?p=1117 and its comments should answer your question. $\endgroup$ – Olivier Benoist May 3 at 12:55
  • $\begingroup$ I have just read that link and I don´t think they come to a conclusion. $\endgroup$ – J. Ove May 3 at 16:26
  • $\begingroup$ Your question 2 is precisely the "wild guess" in the blog post, and BCnrd's comment sketches a proof of it. $\endgroup$ – S. Carnahan May 4 at 1:51
  • $\begingroup$ @OlivierBenoist you’re totally right, Thanks for the reference $\endgroup$ – J. Ove May 4 at 12:52

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