46 votes
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Volume of the unitary group

As indicated by Igor Rivin, the volume of the unitary group is given by $vol(U(N))=(2\pi)^{(N^2+N)/2}/\prod_{k=1}^{N-1} k!$. The denominator is the Barnes G-function, which is well-known : http://...
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  • 6,610
41 votes
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Why not add cuspidal curves in the moduli space of stable curves?

If you add cuspidal curves, then $\overline{\mathcal{M}}_{1,1}$ will no longer be separated, which is the scheme/stack analogue of Hausdorff. Specifically, consider the families $$y_1^2 = x_1^3 + t^6 \...
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36 votes
Accepted

Maryam Mirzakhani's works

A very good expository article (in Farsi) on recent work of Maryam Mirzakhani can be found here. (PDF)
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31 votes
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How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

Riemann combines what is called Riemann-Roch and Riemann-Hurwitz nowadays. He considers the dimension of the space of holomorphic maps of degree $d$ from the Riemann surface of genus $g$ to the sphere....
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30 votes

Why is there no Brauer scheme?

Suppose that $Br(X)$ is representable in the following sense: there exists a $k$-scheme $B_X$ such that for each $k$-scheme $S$ there is a natural bijection $B_X(S)=Br(X_S)$, or perhaps we should ...
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20 votes
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Why there is a Quot-scheme, not a Sub-scheme?

For standard universal properties, you need the scheme to behave well under base change, which in these cases would mean tensor products. Tensor product is right exact, so a quotient remain a quotient,...
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  • 5,542
17 votes
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Which mapping class group representations come from algebraic geometry?

Dan, Although I'm no longer very active on MO, I thought I'd make a few comments, since your question is an interesting one (and you're not anonymous). The paper of Looijenga referenced in Igor's ...
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  • 31.8k
17 votes
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elliptic curves and group cohomology

As Charles indicates, "the moduli stack of $G$-bundles on $E$" is not quite the right thing to consider, especially if you're not working over $\mathbf{C}$. This is for two (unrelated) reasons: 1) ...
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17 votes
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DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons

The Quillen-Drinfeld-Deligne-etc. philosopy should not be looked at as something too mysterious. Namely, it reduces to the fact that if the set of objects one is interesting in the infinitesimal ...
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16 votes
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Special fiber of $X(p)$ in characteristic $p$

A bit of mastication of Katz-Mazur Theorem 13.7.6 and the surrounding text seems to yield the following description of the special fiber of $Y(p)$: It is fundamentally $p+1$ copies of $\mathbb{P}^1$ (...
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  • 43.1k
16 votes

Motivations to study the cohomology of the moduli space of curves

As the title to Mumford's famous paper "Toward an enumerative geometry..." suggests, knowing the cohomology / cycle theory of the moduli space of curves allows one to answer enumerative geometry ...
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  • 5,774
16 votes

Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?

I am answering your "later addon" only, although it seems actually to be a very different question than your original one. This is perhaps one of the most misunderstood aspects of HoTT and ...
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  • 59.4k
15 votes

How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

The original paper of Riemann is his celebrated "Theorie der Abel'schen Functionen" in Crelle's Journal of 1854. This paper can be found online at https://www.maths.tcd.ie/pub/HistMath/People/Riemann/...
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  • 1,178
14 votes

Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?

Here is an answer to your original question in the context of HoTT. An arbitrary map $f:X\to Y$ that isn't a fibration can't be viewed literally as a family of spaces varying continuously over $Y$, ...
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  • 59.4k
14 votes

Moduli space of linear partial differential equations

Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any ...
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  • 24.9k
14 votes
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On the moduli stack of abelian varieties without polarization

First, when defining the stack you will have the issue that there are formal deformations of abelian varieties which do not extend to families of abelian varieties over any reduced scheme. These are ...
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  • 116k
13 votes
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density of singular K3 surfaces

This is a standard argument and there probably exists a reference but it's not hard once you rephrase it in terms of the period domain. The moduli space of K3 surfaces is locally isomorphic to its ...
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  • 116k
12 votes
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Singularities of the moduli stack of Calabi-Yau threefolds

Yes, Calabi-Yau manifolds have unobstructed deformations. This is due to Tian and Todorov; there is a nice algebraic proof in a paper by Kawamata, J. Algebraic Geom. 1 (1992), no. 2, 183–190.
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  • 34.4k
12 votes
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Is Teichmüller distance bigger than Weil-Petersson distance on Teichmüller space?

Yes, this is a result of Michele Linch, 1974.
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  • 93.8k
11 votes
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Koszulness of the cohomology ring of moduli of stable genus zero curves

It is: https://arxiv.org/abs/1902.06318 - this paper also explains how to use the Koszul dual algebra for something, where something is estimating Betti numbers of the free loop spaces of $\overline{M}...
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11 votes

Results about moduli of surfaces

As already said by Simon in his comment, this is a very vast topic. Let us stick for simplicity to the case of smooth surfaces $S$ of general type: in this case it is well known that $h^0(S, \, T_S)=...
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11 votes

Volume of the unitary group

The volume of the unitary groups is standard (I recommend Lando, Zvonkin: Graphs on Surfaces and their applications, Cor. 3.5.2), it is: $$\textrm{vol}~ U_n = \frac{(2\pi)^{(n^2+n)/2}}{\prod_{k=1}^{n-...
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  • 93.8k
11 votes

Axiomatic characterization of virtual fundamental classes?

Question 1 (compare virtual fundamental cycles of different perfect obstruction theories on space underlying space): There is essentially no relation between $[X]_\varphi$ and $[X]_{\varphi'}$ for ...
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  • 17.5k
11 votes
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Prerequisites for reading papers of arithmetic such as Ribet, Mazur, Faltings, Wiles

I don't know what you mean by Modular forms of moduli stack, I think maybe you mean modular forms on moduli stacks. Either way, you should probably have a look at the book by Katz--Mazur titled "...
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11 votes

Moduli space of flat connections of Lie group over a 2-torus

Let $K$ be a connected compact Lie group. The moduli space of flat $K$-bundles over an $n$-torus is homeomorphic to the character variety $Hom(\mathbb{Z}^n,K)/K$. The identity component of this ...
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  • 8,044
11 votes

What is the official definition of $\mathcal{M}_g$ as an orbifold, and how much can I ignore it?

Since you seem to be mainly interested in $\mathcal{M_g}$, let me suggest a "quick and dirty" approach based on the following fact: $\mathcal{M}_g$ is quotient of a nonsingular algebraic variety $\...
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  • 31.8k
11 votes
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Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?

If $X$ is a (smooth projective) curve over $\overline{\mathbb{Q}}$, we define The Belyi degree $\deg_B(X)$ of $X$ to be the minimum degree of a Belyi map $X\to \mathbb{P}^1_{\overline{\mathbb{Q}}}$. ...
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10 votes
Accepted

Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?

Let me elaborate on my comment, adding some details and references. The space $\overline{\mathcal{M}}_{g, \, p}$ is an almost Kähler $V$-manifold. This means that it has only quotient singularities ...
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10 votes

Easiest example where field of definition is not field of moduli

EDIT: This answer is incorrect, for the reason indicated in the other answer; it should be consulted for a correct curve. Here is a recipe for constructing some examples. Suppose that $f$ is a ...
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  • 47.9k

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