# Questions tagged [moduli-spaces]

Given a concrete category C, with objects denoted Obj(C), and an equivalence relation ~ on Obj(C) given by morphisms in C. The moduli set for Obj(C) is the set of equivalence classes with respect to ~; denoted Iso(C). When Iso(C) is an object in the category Top, then the moduli set is called a moduli space.

290
questions with no upvoted or accepted answers

**29**

votes

**0**answers

665 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 ...

**19**

votes

**0**answers

469 views

### Other examples of computations using transfer of structure from the chains to the homology?

There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...

**16**

votes

**0**answers

497 views

### Lie algebras vs. graph complexes

A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...

**16**

votes

**0**answers

2k views

### MNOP conjecture

Let $X$ be a smooth, projective, Calabi-Yau 3-fold (CY makes the exposition more elegant, I don't think it is necessary).
To define Gromov-Witten invariants, we consider moduli spaces of stable ...

**15**

votes

**0**answers

980 views

### Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...

**13**

votes

**0**answers

652 views

### Borel-Weil-Bott, Langlands and Hitchin

Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote by $B$ a Borel subgroup of $G_\mathbb{C}$.
Given a dominant weight $\lambda$, one can construct a line ...

**13**

votes

**0**answers

2k views

### conformal blocks for beginners

I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...

**13**

votes

**0**answers

560 views

### Generalization of Witten's computation of the volume of moduli space

Let $\Sigma$ be a Riemann surface, and let $X:=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$.
There ...

**12**

votes

**0**answers

259 views

### Are Hecke eigenvalues on the cohomology of the Newton polygon strata automorphic?

Fix a genus $g$, a prime $p$, and a Newton polygon $\Delta$ of an abelian variety of genus $g$.
Let $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ be the moduli stack of abelian varieties of genus $...

**12**

votes

**0**answers

767 views

### State of research in moduli space of flat connections

I am a recent PhD student trying to settle into a research topic. Even though I have a current project I am working on, I am not particularly enjoying it and would like to switch. Before braving the ...

**12**

votes

**0**answers

425 views

### Where is the representability of the moduli of curves with framed points proved?

There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of ...

**12**

votes

**0**answers

925 views

### 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 \text{...

**11**

votes

**0**answers

255 views

### What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that $H^{...

**11**

votes

**0**answers

559 views

### gromov witten donaldson thomas correspondence

Let $X$ be a nonsingular projective 3-fold. I am trying to understand the proof of the GW/DT correspondence as presented in Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds. I would ...

**10**

votes

**0**answers

186 views

### Kontsevich's A-infty cohomology classes of the moduli space of curves

In his paper "Feynman diagrams and low-dimensional topology," Kontsevich attaches to each $A_\infty$ algebra a cohomology class (with complex coefficients) on the moduli space of smooth, ...

**10**

votes

**0**answers

378 views

### What is the signficance of the existence of a moduli stack to a moduli problem?

This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.
Let's say we begin with a naive moduli problem: we want a moduli space (whatever space ...

**10**

votes

**0**answers

757 views

### Central Yang-Mills connections, and flat connections with prescribed holonomy

Let $X$ be $\Sigma^g$ which is the Riemann surface of genus $g$, and consider a trivial $G$-bundle over it.
1) In this $2$-d setting, the space of Yang-Mills central connections is the set of ...

**9**

votes

**0**answers

155 views

### Methods to compute the Kodaira dimension of moduli spaces

It is known that the moduli space $\bar{M_g}$ of genus $g$ stable curves over $\mathbb C$ is of general type for $g \geq 24$ with Kodaira dimension $3g-3=\dim \bar{M_g}$.
The idea is that one can ...

**9**

votes

**0**answers

95 views

### Moduli-space interpretation of a morphism of unitary Shimura varieties

Let $G$ be the quasi-split unitary similitude group $GU(2, 1)$, for some choice of imaginary quadratic field $E$; and let $T = GU(1)$ be the torus $Res_{E/Q}\mathbf{G}_m$. Then there's a morphism $\...

**9**

votes

**0**answers

355 views

### The virtual fundamental class as derived intersection

Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...

**9**

votes

**0**answers

520 views

### Roadmap to understanding Gromov's Non-squeezing theorem

I'm a graduate student starting out to venture into the areas of Symplectic Geometry/Topology, and was somewhat motivated by the essence of Gromov's non-squeezing theorem which in a sense made me feel ...

**9**

votes

**0**answers

533 views

### Motivic fundamental group of the moduli space of curves?

Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...

**9**

votes

**0**answers

883 views

### “A theory of generalized Donaldson-Thomas invariants” by Joyce & Song

Is anyone else working through this paper: A theory of generalized Donaldson-Thomas invariants, by Dominic Joyce, Yinan Song? I am trying to verifying example 6.2 (m=2 for simplicity) using only the ...

**8**

votes

**0**answers

233 views

### Boundary of Siegel modular variety

The moduli space of curves has a compactification whose boundary can be understood as the product of moduli spaces of curves of lower genus. Therefore (perhaps naively) one might hope that there ...

**8**

votes

**0**answers

249 views

### Genus=2 theta functions, Arnold's relation, and KZ connection

Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one ...

**8**

votes

**0**answers

184 views

### Fundamental group of moduli space of K3's

According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...

**8**

votes

**0**answers

602 views

### How to calculate the top Chern class of a “functorial” vector bundle on a moduli space of sheaves?

Let $\mathcal M$ be a moduli space of sheaves on a nice variety $X$, with fixed rank $r$ and Chern classes $c_i$, semi-stable with respect to an ample bundle $H$. Let $E$ be a "functorial" vector ...

**8**

votes

**0**answers

552 views

### How do I complete the sketch of proof in 'FGA explained', 5.5.8?

It seems to me that the one part that is difficult to transfer to general coherent sheaves is given only one sentence: "Because we have such a common $m$, we get as before an injective morphism from ...

**8**

votes

**0**answers

480 views

### Easiest way to see Theorem 1.2 of Deligne-Mumford's “The irreducibility of the space of curves of given genus”?

Theorem 1.2 of Deligne-Mumford's 1969 IHÉS paper, "The irreducibility of the space of curves of given genus," is as follows.
If $g \ge 2$ and $C$ is a stable curve of genus $g$ over an ...

**8**

votes

**0**answers

310 views

### Coherent cohomology of the moduli space of curves

Is $H^i\left(\overline{\mathcal M}_g, \mathcal O_{\overline{\mathcal M}_g}\right)$ nontrivial for any $i>0$ and any $g$?
I was not able to find literature on this after searching for a bit, ...

**8**

votes

**0**answers

386 views

### Moduli interpretation of Eisenstein series

Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. ...

**8**

votes

**0**answers

474 views

### Deligne-Mumford moduli spaces and compactification of symmetric matrices

The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered ...

**8**

votes

**0**answers

337 views

### Universal non-degenerate map over the space of complete linear maps

Let $E$ and $F$ be vector spaces and assume for simplicity that $\dim E = \dim F = n$.
Recall that the moduli space of complete linear maps from $E$ to $F$ is the space $M(E,F)$ parameterizing all ...

**7**

votes

**0**answers

170 views

### Moduli space of annuli with marked points satisfying some additional symmetries

Let us consider the space of configurations $\overline{\mathcal{M}}_{0,2,1,(1,1)}$ of an annulus with a marked point on the interior boundary component (let's call it "out") a marked point ...

**7**

votes

**0**answers

144 views

### Geometry of moduli problem in practice: how to check it is connected / irreducible / normal / reduced / locally complete interesection…?

Moduli spaces are very common and useful in the world of algebraic geometry. From the point view of functors, one can already check many geoemtric properties of it. I like examples, and you can assume ...

**7**

votes

**0**answers

274 views

### Stacky proof of no elliptic curves over Z

It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...

**7**

votes

**0**answers

150 views

### Maps between moduli of curves

Let $M_{g,n}$ be the moduli space of $n$-pointed curves, and $M_g[m]$ the moduli space of (unpointed) curves with $m$-level structure.
Fix $m>0$. Is it true that for $n$ large enough, there is a ...

**7**

votes

**0**answers

537 views

### Moduli interpretation of Fargues-Fontaine curve

The Fargues-Fontaine curve is, in his schematic version, a noetherian regular scheme $X$ of dimension 1 associated to a pair $(E,F)$, where $E$ is a local field (i.e. complete w.r.t. a discrete ...

**7**

votes

**0**answers

250 views

### Galois representations associated to the modular tower and automorphy

Consider $\mathcal{M} = \{M_{g,n}, \mu_{g,n}^{g’,n’}\}$, the system of moduli spaces of $n$-pointed smooth algebraic curves along with the basic maps amongst them coming from identifying and ...

**7**

votes

**0**answers

282 views

### What is the kernel of $i^*:H^*(\overline{\mathcal{M}}_{g,n},\mathbb{Q}) \to H^*(\overline{\mathcal{M}}_{g-1,n+2},\mathbb{Q})$?

Let $\overline{\mathcal{M}}_{g,n}$ be the Deligne-Mumford moduli space of stable algebraic curves. I would like to know which rational cohomology classes or at least which tautological classes on $\...

**7**

votes

**0**answers

447 views

### Compactification of the moduli space of Kähler manifolds with negative constant scalar curvatures

Moishezon compactification is very important in the study of the moduli space of varieties which admit canonical metrics. Moishezon showed that any non-projective Moishezon manifold $X$, after a ...

**7**

votes

**0**answers

298 views

### GAGA for moduli problems

In algebraic geometry moduli problems are described by a functor $F:\mathrm{Sch}^{\mathrm{op}}\to \mathrm{Set}$ and it is clear what a solution to a moduli problem is, namely a scheme X such that $F\...

**7**

votes

**0**answers

160 views

### Are rings of modular forms normal?

I am interested in ring-theoretic properties of rings of modular forms. Consider the ring $R$ of integral modular forms for some level, say $\Gamma_1(n)$ -- and to be gentle, let's invert $n$. Algebro-...

**7**

votes

**0**answers

199 views

### For all schemes w/Hilbert polynomial P, exists $m_P$ s.t. no higher cohomology, $I(k)$ generated by globally sections, multiplication is surjective

Consider the following theorem.
For every polynomial $P$, there exists an integer $m_P$ such that for all ideal subsheaves $I \subset \mathcal{O}_{\mathbb{P}^n}$ with Hilbert polynomial $P$ and every ...

**7**

votes

**0**answers

236 views

### Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a non-...

**7**

votes

**0**answers

256 views

### Relations among Hodge classes?

Let $\pi : C_g \to M_g$ be the universal curve over the moduli stack of genus $g$ curves. Let $\omega_\pi$ be the relative canonical bundle. Then $\mathbb{H} := \pi_\ast \omega_\pi$ is a rank $g$ ...

**6**

votes

**0**answers

119 views

### Logarithmic Darboux theorem

Let $X$ be a smooth complex analytic manifold and $D$ be a normal crossing divisor. Suppose that there is a complex analytic logarithmic symplectic structure on $X$.
Is there a Darboux like theorem ...

**6**

votes

**0**answers

98 views

### Presentation of $H^2(\overline{M}_{0,n},\mathbb{Z})$ as an $S_n$-module?

Let $\overline{M}_{0,n}$ be the moduli space of genus zero curves with $n$ marked points. Let $I=\{\{S,S^c\}|S\subset\{1,\dots,n\},|S|\geq2,
|S^c|\geq2\}$ be the set of partitions of $\{1,\dots n\}$ ...

**6**

votes

**0**answers

114 views

### What are the genus 4 curves with Jacobians that are 4-th powers?

Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...

**6**

votes

**0**answers

74 views

### “Moduli space” of isotropic convex bodies?

A lot of questions in convex geometry revolve around the geometry of isotropic convex bodies in $\mathbb{R^n}$.
To my knowledge there is no, or very little study of a space such as :
$$C_n = \{...