41
votes

Accepted

### 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 \...

32
votes

Accepted

### 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....

21
votes

Accepted

### 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,...

18
votes

Accepted

### 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) ...

18
votes

Accepted

### 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 ...

17
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 ...

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 ...

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/...

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 $Y$, ...

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 ...

14
votes

Accepted

### 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 ...

13
votes

Accepted

### 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 ...

12
votes

Accepted

### Is Teichmüller distance bigger than Weil-Petersson distance on Teichmüller space?

Yes, this is a result of Michele Linch, 1974.

12
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 $\...

11
votes

Accepted

### 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}...

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 ...

11
votes

Accepted

### 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 "...

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 ...

11
votes

Accepted

### 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 $X\to \mathbb{P}^1_{\overline{\mathbb{Q}}}$.
...

10
votes

Accepted

### Kontsevich space in positive characteristic

I am just writing my comments as an answer. The main computations have to do with the cotangent complex of a stable map. I will work with unpointed stable maps for simplicity (the associated ...

Community wiki

10
votes

Accepted

### Moduli 'space' of stacks?

Such a moduli problem for stacks is expected to be a $2$-stack.
For example, consider the stack of line bundles on $X$, whose objects are parameterized by $H^1(X, \mathbb{G}_m)$. This is a (trivial) ...

10
votes

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

Since in the comments you seem drawn to the "charts" approach to orbifolds, here are some resources which give definitions and more (there is no "official definition" as far as I know):
Joan Porti ...

10
votes

Accepted

### Moduli of curves over finite field

The constructions of the Deligne–Mumford stack $\mathscr M_g$ and its coarse moduli space $M_g$ are very similar, and Deligne–Mumford's original article [DM69] is surprisingly readable. Note that ...

10
votes

### Moduli of curves over finite field

As requested, I am posting my comment as an answer. Mumford discusses this in the preface to the first edition of "Geometric Invariant Theory".
Already in "Geometric Invariant Theory" (which was ...

Community wiki

10
votes

Accepted

### A "comprehensive" family of abelian varieties

Welcome new contributor. The idea of such "comprehensive" families goes back very far. These were studied by Amitsur under the name "generic splitting varieties", primarily in ...

Community wiki

10
votes

Accepted

### Definition of modular curve associated to $\Gamma(N)$

This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question).
Your $S(N)$ is naturally a scheme over $\mathbb{...

9
votes

### What are some open problems in algebraic geometry?

The Maximal Rank Conjecture is a major outstanding problem in Brill-Noether theory, although recent advances in tropical techniques might point the way to a solution; see https://arxiv.org/abs/1505....

Community wiki

9
votes

### DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons

With Domenico's clear explanation, I can actually write down more or less explicitly the DGLA describing the deformations of Einstein metrics.
First, some notation. Let $\bar{g}_{ab}$ denote a given (...

9
votes

Accepted

### Hyperbolic Metric on a Riemann Surface

Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane
$$
C_R = \{ z \...

9
votes

Accepted

### Homotopy groups of smooth part of moduli space

I'm not sure why you're talking about codimension $1$ components of $S$ since the smooth locus $S$ is actually dense in $\mathcal{M}_g$.
Anyway, the fundamental group of the locus $S$ of smooth ...

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