14

No. In fact more is true: the locus of all $n$-pointed curves of genus $g-1$ with a single elliptic tail $E$, such that $\mathrm{Aut}(E)=\mathbf Z/6$, has codimension two in $\overline M_{g,n}$ and consists of noncanonical singularities. This was famously determined by Harris and Mumford in their paper on the Kodaira dimension of the moduli space of curves (...

12

Often you hear the informal statement that a stack is like a scheme, except with a stabilizer group attached to each point. Your question shows why the intuition from this statement can be misleading. In a sense, you need to also remember the way these groups fit together. But if you try to make that statement precise you'll soon find yourself completely ...

11

Except in a few trivial cases, the locus of curves which have an extra automorphism will have codimension greater than one in $\mathcal M_g$. When that happens, the fundamental group of $\mathcal M_g$ must equal the fundamental group of $\mathcal M_g^{\circ}$. (To see this, one can pass to the $3$-torsion cover of $\mathcal M_g$, which is a smooth scheme, ...

11

Since $\mathcal{M}_g=[\mathcal{T}_g/\text{Mod}_g]$, and $\mathcal{T}_g$ Teichmueller space is contractible, (1) and (3) are going to be isomorphic.
This was mentioned previously on MathOverflow here, or can be found for instance on page 361 of Farb and Margalit "A primer on Mapping Class Groups".
Meanwhile, (2), the fundamental group of the coarse moduli ...

8

Yes, this would imply that $\newcommand{\X}{\mathcal X}\X^s$ is the coarse moduli space, but I don't think this is the "right" question to ask -- I believe that $\X^s$ will not even form a sheaf unless $\X$ happens to be a scheme/algebraic space to begin with.
Anyway, any morphism from a groupoid to a set factors through $\pi_0$ of the groupoid. This ...

8

$\overline{\mathfrak{M}}_{0,0}$, the stack of pre-stable genus zero curves. This parameterizes genus zero curves having at worst nodal singularities. There are an infinite number of isomorphism classes of such curves, but they all occur as a specialization of a trivial family (just do repeated blowups of the central fiber of the family $\mathbb{P}^1\times \...

8

I wanted to work this out for myself anyways, so here's a summary of the argument in Le Potier.
Suppose $X$ is a projective variety, and $E,G$ are locally free sheaves on $X$. Put $S = {\rm{Ext}}^1(G,E)$, and let ${\bf E},{\bf G}$ be the constant families on $S\times X$, namely ${\bf E} = p_2^\ast E$ and ${\bf G} = p_2^\ast G$. We want to construct a ...

8

The GIT proof gives very nice compactifications of these spaces (and is the "right" way to do this), but they were known to be quasiprojective varieties long before GIT was developed.
The classical proofs depend on properties of theta functions. For $\mathcal{A}_g$, it should be attributed to some combination of Satake and Baily, and the appropriate ...

8

One can see the difference by writing down the functor of points explicitly. For a test scheme $T$,
\begin{align}
(\mathbb G_m\times BH)(T) & =\{(f,p)\mid f:T\to \mathbb G_m, p:T'\to T\text{ $H$-torsor}\},\\
[\mathbb G_m/G](T) & =\{(q,g)\mid q:\tilde T\to T\text{ $G$-torsor, }g:\tilde T\to \mathbb G_m\text{ $G$-equivariant}\}\\
& =\{(f,p')\mid f:...

7

I am just adding some details to my comments above. Let $k$ be a field. Let $G$ be a simply connected, semisimple algebraic $k$-group; for simplicity, assume that $G$ is split. Let $X$ be a projective $k$-scheme with a transitive, smooth action of $G$. For simplicity, assume that $X$ has a $k$-point with stabilizer $P$ a ("standard", i.e., smooth) ...

6

In the "more sophisticated" direction, we can ask a similar question about the moduli stack $\mathscr{M}_g$ of hyperelliptic curves of genus $g$. If $K$ is a topological field, there is a natural topology on the set $\vert\mathscr{M}_g(K)\vert$ of isomorphism classes of of genus $g$ hyperelliptic curves over $K$: a subset $\Omega$ of $\vert\mathscr{...

answered Oct 29 '20 at 9:32

Laurent Moret-Bailly

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6

Here is a classifying problem without a coarse moduli space: Let $F$ be the stack of line bundles with section, say over an algebraically closed field k. That is $F(X)$ is the category of pairs $(L,s)$ where $L$ is a line bundle on $X$ and $s \in \Gamma(X,L)$ is a section of $L$. Then $F$ has no coarse moduli space: if it did, there would be two points, ...

5

I don't know if this is exactly the answer you are looking for but for even $n$ this is clearly related to the result by Katsylo for moduli spaces classical binary forms or hyperelliptic curves, see this paper.
Another link for accessing the article by Katsylo is this one.
Theorem 0.2 therein seems to give a positive answer to your rationality question for ...

answered Dec 4 '14 at 15:14

Abdelmalek Abdesselam

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4

There is an example in Harris-Morrison Moduli Of Curves (Exercise 1.7): the moduli functor $F:\mathfrak{Sch}/\mathbb C\to\textrm{Sets}$ that sends a scheme $B$ to the set of isomorphism classes of (flat $B$-families of) reduced plane curves of degree $2$. The point is to show that there exists a "universal" natural tranformation $\eta:F\to \hom_{\mathbb C}(-,...

4

The moduli space of semi-stable vector bundles with trivial determinant over a genus $g$ curve. If the rank is 2 then the coarse space is isomorphic to $\mathbb{P}^3$!!

4

I do not believe the result about stability of tricanonical curves is a formal consequence of asymptotic stability. Rather, I think that the explicit arguments that prove $m$-stability for $m\gg 0$, in fact, already apply if m ≥ 3 if $m\geq 5$. The standard reference is the following.
MR0450272 (56 #8568) Reviewed
Mumford, David
Stability of ...

4

The two notions are related using Theorems 4.1 and 6.1 of the paper of Brosnan, Reichstein and Vistoli:
Theorem 6.1 reduces the computation of the essential dimension of the stack to that of the generic gerbe $\mathcal{X}_g$. Theorem 4.1 says that the essential dimension of $\mathcal{X}_g$ is $\mathrm{cd}(\mathcal{Y}_g) - 1$, where $\mathrm{cd}(\mathcal{Y}...

4

The moduli space $\overline{M}_{0,6}$ can be realized by blowing-up in $\mathbb{P}^3$ five points $p_i$ and then the strict transforms of the ten lines $l_{i,j} = \left\langle p_i,p_j\right\rangle$ through two of them. This is the Kapranov's construction of $\overline{M}_{0,n}$.
Now, consider in $\mathbb{P}^3$ the unique smooth quadric surface $Q$ ...

4

If $n\geq 5$ then $Aut(\overline{M}_{0,n})\cong S_n$ (http://arxiv.org/abs/1006.0987). For instance $\overline{M}_{0,5}$ is a Del Pezzo surface of degree five. Its automorphism group is well known to be $S_5$ (http://arxiv.org/abs/math/0610595).
In particular, any automorphism of $\overline{M}_{0,5}$ preserves the boundary.

4

One modular interpretation of $X_1$ mimics the modular interpretation of the space of complete collineations. Let $V$ be a $k$-vector space of finite dimension.
Definition 1. For every $k$-scheme $T$, for every invertible $\mathcal{O}_T$-module $\mathcal{L}$, a $(V,\mathcal{L})$-system on $T$ is a homomorphism of $\mathcal{O}_T$-modules, $$\phi:V\...

3

A more detailed description of the singular locus of $\mathrm{M}_g$ is as follows.
Theorem. Let $\mathrm{C}$ be a smooth curve of genus $g$.
If $g=2$, then $[\mathrm{C}]$ is a singular point of $\mathrm{M}_2$ if and only if $\mathrm{C}$ is given by $y^2=x^6-x$.
If $g=3$ and $\mathrm{C}$ is not hyperelliptic (resp. hyperelliptic), then $[\mathrm{C}]$ is ...

3

I will just develop a bit the comment by Jason Starr. In these notes
http://math.stanford.edu/~conrad/papers/coarsespace.pdf
you can find the following theorem by Keel and Mori:
Let $S$ be a scheme and let $\mathcal{X}$ be an Artin stack that is locally of finite presentation over $S$ and has ﬁnite inertia stack $I_S(\mathcal{X})$. There exists a coarse ...

3

You should read Arbarello-Cornalba more carefully! Let's consider the gluing map
$$ h:\overline M_{g,n+1} \times \overline M_{g',n'+1} \to \overline M_{g+g',n+n'}.$$
As you say we have $H^2(\overline M_{g,n+1} \times \overline M_{g',n'+1}) \cong H^2(\overline M_{g,n+1}) \oplus H^2(\overline M_{g',n'+1})$ by the Kunneth theorem, since both factors have ...

3

Here is a partial answer to your question. If $X$ is homogeneous then $\overline{M}_{0,0}(X,\beta)$ is a projective normal variety with at most finite quotient singularities. The singularities arise along the loci parametrizing maps with non trivial automorphisms. However, if such a locus is in codimension one the general point of the divisor is a smooth ...

2

We denote by $\Delta_{1}$ the image in $\overline{M}_{g,n}$ of $\overline{M}_{1,1}\times \overline{M}_{g-1,n+1}$, the divisor parametrizing curves with elliptic tails.
Let $Z_6$ the image of $[E_6]\times \overline{M}_{g-1,n+1}\subset \Delta_{1}$, the codimension two loci where the elliptic tail has six automorphisms.
For any $g\geq 1$ the sub-variety $Z_6$...

2

This answer is basically a summary of the comments above. First, for $N$ equals $1$ and for $d\geq 2$, for every integer $n$, the morphism is not birational. One can analyze the pushforward of the structure sheaf; there is something about this in "The Kodaira dimension of spaces of rational curves on low degree hypersurfaces".
For $N$ equals $2$, denote ...

2

In that Deligne-Mumford's paper they proved stability for m≥5*. To consider the pluricanonical embedding $\omega_C^m$ with $m=3$ gives a different compactification of $M_g$. In particular cusps are stable and elliptic tails are unstable. This case was worked out by Schubert (a Gieseker student). The article is: A new compactification of the moduli space of ...

2

I started writing an answer about the generators and relations for the $\mathbb{Q}$-Picard group of the stack of stable maps of genus $0$ curves to an arbitrary projective homogeneous variety, but it quickly got too long. So here is an explanation of the extra divisor class relation that I was missing in my comments. This uses Lemma 5.2 of my article with ...

2

I am posting this as an answer because the comment thread is too long. A proper morphism of integral, Noetherian schemes, $f:X\to Y$, is a contraction if the sheaf homomorphism $f^\#:\mathcal{O}_Y\to f_*\mathcal{O}_X$ is an isomorphism. Typically we only talk about this when $X$, or at least $Y$, is normal, because then we have Zariski's Main Theorem and ...

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