I was wondering if there was a really good place to learn about this topic that actually contains the details of the construction from beginning to end. I'm familiar with the book by Harris and Morrison, and many of the papers about the construction. The recent book by Arbarello, Cornalba, and Griffiths covers the complex case in a very nice and detailed way, but I was wondering if there was a good place to learn this 1) that only uses algebraic geometry and not complex analytic geometry, and 2) that covers the construction over $\mathbb Z$ so one can use it in the arithmetic case. Most books simply leave so many details out that a beginner in the subject finds it hard to fill them all in and truly understand the construction. Any good references would be much appreciated. Thanks

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    $\begingroup$ OMG! Volume II of "Geometry of Algebraic Curves"?! I'm so happy that it's finally coming out -- I had actually given up all hope that it would ever happen. $\endgroup$ – Kevin H. Lin May 25 '11 at 8:24
  • $\begingroup$ a link to the book: springerlink.com/content/… $\endgroup$ – quim May 25 '11 at 9:03
  • $\begingroup$ The coarse moduli space over $\mathbb{Z}$ is constructed in Mumford's Geometric Invariant Theory. An easier to read reference might be the article by Mumford: Stability of projective varieties. Enseignement Math. (2) 23 (1977), no. 1-2, 39–110. As a stack, it is constructed in the paper of Deligne and Mumford "The irreducibility of the space of curves of a given genus". In fact, they construct the moduli stack of stable curves over $\mathbb{Z}$. $\endgroup$ – naf May 25 '11 at 12:38
  • $\begingroup$ @ulrich: I have Mumford's GIT, but does he actually include the details there? In the famous paper of DM, already on the first page they start quoting details from Hartshorne's Duality and Residues, which I haven't studied. I was hoping for something more beginner friendly, which included the details of why the relative dualizing sheaf is ample and in fact why is exists/what it is. I know now how to describe it explicitly but only after piecing together many different sources. Is there anything which actually explains the machinery necessary? $\endgroup$ – HNuer May 25 '11 at 15:49
  • $\begingroup$ I don't know any reference which explains all the machinery. Another good reference is Gieseker's "Lectures on moduli of curves" which gives a fairly complete algebraic construction of the moduli of stable curves using GIT. $\endgroup$ – naf May 27 '11 at 12:43

These notes by Dan Edidin: http://www.math.missouri.edu/~edidin/Papers/mfile.pdf give a very readable account of the construction of the moduli stack of curves over $\mathrm{Spec}(\mathbb Z)$ for $g \geq 3$ via the Hilbert scheme. I am not sure if this is what you are looking for, though, depending on what you mean with "from beginning to end": certainly there are several steps left as pointers to the literature.

  • $\begingroup$ I'm familiar with Edidin's notes which are good, but is there anything more explicit than that? $\endgroup$ – HNuer May 25 '11 at 15:52

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