Talk:Moduli of algebraic curves

Rename article

The current name is unreasonably long. "Moduli of stable curves" or "Moduli stack of stable curves" would be more in line with normal usage — the fact that the stack is a Deligne-Mumford stack is not normally incorporated into its name. I in fact prefer the first one, as it better accommodates the discussion of the coarse moduli scheme as well (separate articles for these would not be a good idea). Stca74 (talk) 16:05, 5 August 2008 (UTC)

I don't think any of the currently proposed names are acceptable, not the present name nor either of your replacements. What I think we should instead have, and what I think best represents the content of the current article, is an article named "Moduli space of curves" which describes M_g (the open part) as well as its various compactifications (of which the Deligne-Mumford compactification is only one). If there is a need for a separate article on the Deligne-Mumford compactification then it should be titled "Deligne-Mumford compactification of the moduli space of curves". Ozob (talk) 19:58, 6 August 2008 (UTC)

By the way, I think it's worth pointing out that there is already an article on stable maps, though it discusses the symplectic case exclusively. Ozob (talk) 20:02, 6 August 2008 (UTC)

Your proposed name would be fine, although I would then rather have it in the plural: Moduli spaces of curves, as it would most naturally discuss several variants (coarse moduli schemes, the fine moduli stacks (sometimes schemes), the locus of smooth curves a well as the completed case of stable curves, cases with and without marked points. I suppose your objection to the name Moduli of stable curves is that you would not want too many articles on the various spaces? I agree with this logic. Otherwise I cannot see why my proposals would be unacceptable, even considering the extremely thin content in the current article (which is less useful or complete than the section of moduli space that it is supposed to be an expansion of, the lead of which is really about the Deligne-Mumford classic article (and where the primary stack defined is the stack of stable curves). Stca74 (talk) 06:14, 7 August 2008 (UTC)

I'd rather have it in the singular, since I think that usage is more common and we're not supposed to be inventing new terminology on Wikipedia. But there ought to be a redirect from the plural, and I could be convinced that the plural is more appropriate.

You figured out exactly why I object to "Moduli of stable curves". Maybe someday, Joe Harris will find this page and make it so rich with content that we need to distinguish facts about M_g from facts about the Deligne-Mumford compactification. But we're nowhere near there yet. Ozob (talk) 20:32, 7 August 2008 (UTC)

This is getting interestingly iterative... I see your point. However, rather than proposing new terminology, my point was that, as the eventual article should discuss the several different variants of moduli spaces of curves, having the article title in the plural would be appropriate. After all, with the intruduction of marked curves we expand from simply compactifying another moduli space. Now, on the other hand, a quick unsystematic review of WP article naming practice finds very few articles with title in the plural - these are almost all redirects to the same in the singular.

Thinking about this more (and given that we seem to agree that the eventual article should be rather inclusive until the (distant) time when there is so much material that it should be divided), I would gravitate towards a modification of my original proposal: how about Moduli of curves or Moduli of algebraic curves, in the tradition of the Katz-Mazur book on moduli of elliptic curves. After all, there is more to a moduli problem than just the space.Stca74 (talk) 12:20, 8 August 2008 (UTC)

Sure, either of those would work for me. (I did think about suggesting "moduli space of algebraic curves", since I think it's a little more precise (one could imagine construing curve in the non-algebraic sense), but I think it's more common to leave out "algebraic".) It's your pick, since I don't think there's a lot to distinguish the two. Ozob (talk) 22:56, 8 August 2008 (UTC)

Completed the move, choosing the latter version to distinguish from non-algebraic moduli problems (mainly moduli of J-holomorphic curves). Updated the importance to High to reflect the wider scope of the intended article. The currently existing article is truly just a stub, and as a first quick fix I'll copy material from the section Moduli space#Moduli of curves, for which this article should become the main page. Stca74 (talk) 18:07, 9 August 2008 (UTC)

Additional references

• http://www.numdam.org/item/AST_1988__165__1_0/ (last section on genus 3 curves)
• https://link.springer.com/article/10.1007/BF03031436
• https://mathoverflow.net/questions/76585/moduli-space-of-genus-2-curves
• http://www.dam.brown.edu/people/mumford/alg_geom/papers/1972d--IntroModuli-NC.pdf
• Quasi-projectivity of M_g https://www.jstor.org/stable/pdf/1970178.pdf?seq=1
• General type of g \geq 23 - The Kodaira dimension of moduli of curves - https://link.springer.com/article/10.1007%2FBF01388710
• http://fse.studenttheses.ub.rug.nl/9712/1/Martijn_van_der_Valk_WM_2011.pdf — Preceding unsigned comment added by Wundzer (talk • contribs) 00:56, 8 June 2020 (UTC)

Stratification of M_g

There is a stratification of the smooth locus of ${\mathcal {M}}_{g}$ given by the Weierstrauss locus. The details are in http://www.numdam.org/item/CM_1974__29_3_325_0/ — Preceding unsigned comment added by Wundzer (talk • contribs) 23:05, 7 June 2020 (UTC)

Compactifications

• Stable compactification
• CM compactification
• https://arxiv.org/abs/1012.0329 -> survey
• https://arxiv.org/abs/0808.0177

Construction of Mg,n

• https://www.mscand.dk/article/view/12001/10017 -> overview
• https://eudml.org/doc/166839 -> construction with algebraic stacks over $Z$
• Sketch from notes by conrad -> http://virtualmath1.stanford.edu/~conrad/249BW17Page/handouts/univfamily.pdf
• http://virtualmath1.stanford.edu/~conrad/249BW17Page/handouts/alterations-notes.pdf -> apparently constructing this stack directly is hard..
• Geometry of Algebraic Curves II pages 257, 268, 274, 279, 294, - 323

Geometry of algebraic curves II

• section 2 constructs a normal analytic space for M_g,n
• section 3 constructs an associated algebraic space
• section 5 has a construction with a special Hilbert scheme
• section 9 has universal curves
• section 10 has properties of moduli, such as clutching maps

Special Hilbert Scheme

• v-log-canonically embedded n-pointed curves (page 193 onward)
• Page 170 has overview + definition in this space, 196 has defubutuib
• Log-canonical sheaf: 92, 99, 195,