The moduli-spaces tag has no wiki summary.

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### Arithmetic and moduli spaces of higher genus curves

Modular curves (as moduli of elliptic curves with level structure) play a key role in the study of the arithmetic of elliptic curves. The higher genus curves have very different arithmetic, but I ...

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### What is $M_g$ over a finite field, really?

Let $M_{g} \to \mathbb{Z}$ be the coarse moduli scheme of non-singular genus g curves over $\mathbb{Z}$. That is, suppose that $M_{g}$ co-represents the functor $M^{\sharp}_{g} : \text{Sch} \to ...

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**1**answer

376 views

### Picard group of $\mathcal{M}_{0,n}$

Let $\mathcal{M}_{0,n}$ be the complement of the boundary of the Mumford-Knudsen compactification of the moduli space of genus zero, n-pointed curves.
Is $Pic(\mathcal{M}_{0,n})$ trivial?

**2**

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**1**answer

265 views

### Differntial of the Torelli morphism and the multiplication map

Consider the Torelli morphism $T$: $M_{g}$-----------> $A_{g}$. The differential of this morphism is the map
$dT$ : $ T_{M_{g}}$---------> $T_{A_{g}}$
between the tangent bundles. Now at the point ...

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**4**answers

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### Curves which are not covers of P^1 with four branch points

The following interesting question came up in a discussion I was having with Alex Wright.
Suppose given a branched cover C -> P^1 with four branch points. It's not hard to see that the field of ...

**16**

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**1**answer

545 views

### Do mapping classes have gonality?

(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.)
The hyperelliptic mapping class group is (by ...

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**3**answers

450 views

### Tabulation of known unstable rational homology of moduli space?

Does there exist a tabulation of the known rational homology of mapping class groups of genus $g$ with $m$ punctures? I'm most interested in the case when punctures can be permuted.

**4**

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412 views

### A question about moduli spaces over $\mathbb{Z}$

I've seen, on several occasions, papers whose purpose it is to construct a moduli space over $\mathbb{Z}$ for a moduli problem for which a moduli space over $\mathbb{C}$ was already constructed. Let's ...

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**2**answers

425 views

### More questions about log structures

I had previously asked:
Help motivating log-structures
I now have some more questions regarding the role of log structures in moduli problems (you can assume that the moduli problem is the ...

**22**

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**3**answers

1k views

### Help motivating log-structures

I'm currently reading a thesis that uses log-structures. I should mention that this is my first encounter with them, and the thesis (as well as my expertise) is scheme-theoretic (in fact ...

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**1**answer

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### Is there an algebraic description of Yang-Mills measure?

Consider $X=\operatorname{Hom}(\pi,SL(2,\mathbb C))/\\!/SL(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,SU(2))/\\!/SU(2)$, where $\pi$ is a surface group. Note that if we use the right coordinates, ...

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### Why is the degree:rank ratio of a vector bundle called its “slope”?

Whenever one studies moduli spaces of vector bundles on curves, one of the first things to be introduced is the "slope" of a vector bundle, i.e., its degree:rank ratio. Is there a nice (preferably ...

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**2**answers

459 views

### Do coarse moduli spaces respect Galois actions?

To explain, I will use the following concrete example: Let $\mathcal{M}_g$ be the functor for the moduli problem of classifying genus $g$ smooth projective curves (taking a scheme $S$ to the set of ...

**6**

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**1**answer

283 views

### Some questions on moduli of stable maps

Let $\overline{M}_{0,k}(\mathbb{P}^n,d)$
denote the moduli space of genus zero degree $d$ stable maps with $k$ marked points. This is an orbifold of expected dimension. Let ...

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**2**answers

446 views

### Has anyone studied the Prym map for double covers with two ramification points?

If $f \colon C \to C'$ is a dominant morphism of smooth projective curves, there is a norm map $f_\ast = \mathrm{Nm} \colon JC \to JC'$ between their Jacobians, and we can consider the abelian ...

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**2**answers

410 views

### A convex polyhedral analog of the pentagram map

I am wondering if there is a three-dimensional analog of
the pentagram map, which maps a convex polygon to another
convex polygon. Here's the Wikipedia image:
...

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**3**answers

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### Is there an algebraic construction of the Quillen (determinant) Line Bundle?

Let's consider the moduli space of representations of $\pi=\pi_1(\Sigma)$ (a surface group) into $G$ (a lie group). Call this $X=\operatorname{Hom}(\pi,G)$, and let ...

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954 views

### Groupoids vs Pseudogroups

(Warning: I'm not an expert in the topic) Let's work in a "geometric" category, for example the category $\mathfrak{Diff}$ of "manifolds" (without the requirements of connectedness and second ...

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**1**answer

402 views

### When does a Shimura variety have contractible universal cover?

Disclaimer: I know very little about Shimura varieties.
Some Shimura varieties have a contractible universal covering space, for instance $A_g$ itself. Are there any nice necessary and/or sufficient ...

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**0**answers

530 views

### level structures and moduli of abelian varieties

Hello,
In the definition of level structure of level $n$ for an elliptic curve $A$, there are two versions:
an isomorphism of group schemes $(\mathbf Z/n\mathbf Z)^2 \to A[n]$.
an isomorhpism of ...

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468 views

### Why do H_4 and M_4 have the same virtual Euler characteristic?

Here's a funny coincidence:
The virtual (or "orbifold") Euler characteristic of $\mathcal M_g$ is known by the work of Harer and Zagier: one has $\chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)$.
Now ...

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**2**answers

440 views

### Generalized Quot-schemes

Given $S=\mathbb{P}^2$ and a locally free $O_S$-module $E$ of rank r and an integer $l\geq 1$. Then it is known that the scheme $Quot(E,l)$ is irreducible, due to Ellingsrud and Lehn. Here $Quot(E,l)$ ...

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212 views

### Projectivized Normal Cone to Satake Compactification

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties over $\mathbb{C}$.
There exists a compactification, the Satake compactification, which is minimal and has the ...

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**1**answer

300 views

### Pullback along the Torelli map is an isomorphism

I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard ...

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**1**answer

244 views

### notion of stability in a category

This is a question in general sense, but answers about specific examples are also welcome.
Why do we need the notion of stability of objects in a category. If we've a subcategory of stable objects, ...

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**1**answer

496 views

### Normality of a locus of points in projective space

Let $U_{d,n}\subseteq(\mathbb{P}^d)^n$ denote the locus of $n$-distinct points in projective space $\mathbb{P}^d$ that lie on a rational normal curve of degree $d$, and let $V_{d,n}$ denote its ...

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494 views

### Does a degeneration always have a larger-dimensional automorphism group?

Suppose $\newcommand{\X}{\mathcal{X}}\X$ is an algebraic stack over a field $k$, $\xi$ is a $k$-point which has another $k$-point $x$ in its closure ($x$ is an isotrivial degeneration of $\xi$). ...

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**1**answer

563 views

### Flat Principal Connections and Homotopy Groups?

I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...

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**1**answer

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### Conformal blocks vector bundles on $\overline{M}_{g}$ in terms of generalized theta functions?

Conformal field theory uses representation theory to produce various vector bundles on the Deligne-Mumford compactified moduli spaces $\overline{M}_{g}$ and $\overline{M}_{g,n}$, known as bundles of ...

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**1**answer

626 views

### Basic Questions about Teichmuller's theorem/quadratic differentials

I have some basic questions about Teichmuller's theorem, since I am a beginner, my questions might be very basic. If you can give some hints/answers or cite some references to study from, I will ...

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660 views

### Are representations of a linearly reductive group discretely parameterized?

Suppose $G$ is a linearly reductive group over a field (say $\mathbb C$). Does somebody know of a proof that any flat family of finite-dimensional representations of $G$ must be locally constant?
In ...

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**1**answer

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### Proposition 3.93 of Harris-Morrison (rational classes on Deligne-Mumford moduli stack vs. rational classes on Deligne-Mumford moduli space)

Proposition 3.93 of Harris-Morrison's "Moduli of Curves" describes the relationship between rational divisor classes on the Deligne-Mumford moduli stack and rational divisor classes on the ...

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### Modular interpretation of an action of the linear group SL_2 on the cohomology of an elliptic curve

Let $E$ be an elliptic curve and $x,y \in H^1(E, \mathbb{Q})$ be a basis for the first rational cohomology group of $E$. There is an action of the linear group $SL_2(\mathbb{Q})$ on ...

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### How were moduli spaces defined before functors?

People today in algebraic geometry will typically define a moduli space to be the space which represents the functor of families of whatever object they are interested in studying.
However, I am ...

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553 views

### Can a rational family of genus-g curves have generic gonality? Can it be Brill-Noether general?

We know that M_g is general type for g large enough. In particular, the generic genus-g curve is not contained in a (non-isotrivial) rational family parametrized by P^1. In fact, the high-genus ...

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### Given a family of curves, when does there exist a fibered surface over Spec Z parametrizing them?

Let $X_p$ be a projective curve over the finite field $\mathbf{F}_p$ (i.e. a projective $\mathbf{F}_p$-scheme pure of dimension 1) for every prime number $p$. Let $X_\mathbf{Q}$ be a projective curve ...

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636 views

### Upper bounds for ranks of modular jacobians

The following question came to me earlier as a "side question"; something I'd like to know, but which is not totally necessary for what I'm thinking about or doing:
Consider the genus 32 curve ...

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**1**answer

1k views

### Which Riemann surfaces arise from the Riemann existence theorem?

The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi_1(X \backslash \Delta) \to S_d$ ...

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461 views

### Is there Harer stability for moduli of curves with level structure?

The famous Harer stability theorem asserts that the homology group $H_d(\mathcal{M}_{g,n},\mathbf{Z})$ is independent of g and n in the range $0 \leq 2d < g-1$. This is proven by analyzing the maps ...

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**1**answer

352 views

### “extend a functor”

Hi,
I have probably a basic question. I have a functor $F: Sch \rightarrow Set$, an algebraic stack $M$ with a "universal family" $G\rightarrow M$ and a representability property like this: for every ...

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**4**answers

884 views

### Why does (Ribbon) Graph (co)Homology Compute (co)Homology of MCG?

The title says it all. I am looking for an explanation or reference for why the homology of the ribbon graph complex computes the cohomology of the mapping class groups of surfaces.
I've seen ...

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**1**answer

204 views

### What is the closure of product loci in A_g?

Let $A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$. For any partition of $g$, one can consider the corresponding locus inside $A_g$ of products of ...

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473 views

### What are the relative differential forms of a family of (nodal) curves?

What is the definition of relative differential forms of a family $\pi: X \to B$ of (nodal) curves, where $B$ is the base space.

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365 views

### Choosing tau for elliptic curves over the rational numbers with prescribed ramification data

Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. Let $B$ be the divisor $$B:= \sum [b_i].$$ We consider this data to be fixed. For $d>1$, we define ...

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586 views

### When can Witten-esque moduli spaces be used to define invariants of geometric structures?

I am trying to understand the big picture around Seiberg-Witten invariants of 4-manifolds. Of course, this points to Taubes work on Gromov-Witten invariants of symplectic manifolds.
It is striking ...

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**3**answers

613 views

### Are there graph models for other moduli spaces?

Recall that a ribbon graph is a graph with a cyclic ordering at each vertex and such that each vertex has valence greater than or equal to 3. This cyclic ordering exactly gives one the information to ...

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681 views

### Proving that a map is a morphism

Example : Consider the (open, not compactified) moduli space of stable maps $ \mathcal M_g(1,d)$ of maps of smooth curves of genus $ g$ to $\mathbb P^1$. To each map
we associate its branch divisor, ...

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299 views

### Quotient of an abelian surface by a finite group, irreducible components

Given an abelian surface $A$ as the product of two non isogenous elliptic curves $E_1 $ and $E_2$ over $\mathbb{C}$. We also have a smooth 2 dimensional moduli space $M$ of sheaves on $A$ with some ...

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### Cohomology of the Moduli of G-bundles on a Curve

For a simple complex group G and Riemann surface X, are the (integral, if possible) cohomology groups of the moduli of holomorphic G-bundles on X written down somewhere, either explicitly or ...

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### Reference for the Hodge Bundle

For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to ...