**25**

votes

**0**answers

511 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 ...

**17**

votes

**0**answers

374 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). ...

**12**

votes

**0**answers

382 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 ...

**11**

votes

**0**answers

205 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 ...

**11**

votes

**0**answers

428 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 ...

**11**

votes

**0**answers

850 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 ...

**11**

votes

**0**answers

1k 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 ...

**10**

votes

**0**answers

387 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 ...

**10**

votes

**0**answers

348 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 $\pi_1(\Sigma)$. ...

**10**

votes

**0**answers

702 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 ...

**9**

votes

**0**answers

349 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 ...

**9**

votes

**0**answers

346 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 ...

**8**

votes

**0**answers

242 views

### A question about Mirzakhani et. al.'s algebraicity theorem

While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...

**8**

votes

**0**answers

285 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 ...

**7**

votes

**0**answers

525 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 ...

**7**

votes

**0**answers

652 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 ...

**6**

votes

**0**answers

151 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 ...

**6**

votes

**0**answers

213 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. ...

**6**

votes

**0**answers

167 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 $g$ ...

**5**

votes

**0**answers

143 views

### Is the moduli space of curves arising from wild ramification smooth?

Fix a natural number $g$, a prime $p$, and a $p$-group $P$.
Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...

**5**

votes

**0**answers

197 views

### Topology of the space of foliations on a 3-manifold

Denote by $\mathcal{P} (M)$ the space of smooth plane fields(oriented and transversely oriented) on a given closed and orientable 3-manifold $M$ with the $C^{\infty}$ topology, and by $\mathcal{F}(M)$ ...

**5**

votes

**0**answers

128 views

### Compactifying the space of indecomposable abelian varieties

Let $A_g$ be the moduli space of principally polarized abelian varieties and $A_g^0$ the open substack of indecomposable ones. Abstractly we know $A_g^0$ has a compactification with complement a ...

**5**

votes

**0**answers

182 views

### Is the moduli space of genus three smooth quartics affine?

Non-hyperelliptic curves of genus three are smooth quartics. Is the moduli space of such curves affine?
I think this follows from a more general result on smooth complete intersections, but I'm ...

**5**

votes

**0**answers

235 views

### Universal nondegenrate 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 ...

**5**

votes

**0**answers

593 views

### Curious propositon in “Les schemas de modules de courbes elliptiques”

Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation):
(II ...

**4**

votes

**0**answers

138 views

### Models for the moduli space $\overline{M}_{1,n}$

Let $\overline{M}_{1,n}$ denote the coarse moduli space of $n$-pointed elliptic curves. Is there an explicit description of these spaces (a la Kapranov's construction) for low $n$?
Apparently this ...

**4**

votes

**0**answers

165 views

### Quadrics and Moduli Spaces

It is well known that $\overline{M}_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret ...

**4**

votes

**0**answers

152 views

### Moduli interpretation of Hecke operators on Shimura curves

In his book on Automorphic Forms, Shimura gives (chapter 9) definitions of the the Hecke operators for Shimura curves.
One can give definitions of the Hecke and Atkin-Lehner operators in terms of the ...

**4**

votes

**0**answers

194 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 ...

**4**

votes

**0**answers

114 views

### Deformations and moduli of semistable sheaves in mixed characteristic

Let $X$ be a projective scheme over an algebraically closed field $k$. There is the coarse moduli space $M_X$ parametrizing semistable sheaves on $X$ with fixed reduced Hilbert polynomial $p$. Now, ...

**4**

votes

**0**answers

136 views

### Shimura varieties and Maximal conditions

Working with Shimura varieties, I have been convinced to call them (or the families giving rise to them especially in $A_{g}$) somehow the "maximal" families. The motivation of this, has been for ...

**4**

votes

**0**answers

255 views

### Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...

**4**

votes

**0**answers

58 views

### Moduli space of CY3 with $h^{2,1}=1$ is $\mathbb{P}^1\setminus \{p_i\}_i$?

It seems to me that all known CY3 with $h^{2,1}=1$ has the complex moduli space of the form $\mathbb{P}^1\setminus \{p_i\}_i$ for some $i \in \mathbb{N}$.
Is this true for all CY3 with $h^{2,1}=1$? ...

**4**

votes

**0**answers

353 views

### Possible automorphism groups of a K3 surface

Which finite groups are automorphism groups of polarized K3 surfaces (let's say over ℂ)?
...and does the answer change is I remove "polarized"?
(polarized = equipped with an ample line bundle)

**4**

votes

**0**answers

191 views

### stability of parabolic bundles beyond the Weyl alcove and Hecke transforms

When considering (quasi)parabolic vector bundles over a smooth complete curve $X$ with marked points $p_i$, it goes back to the foundational work of Seshadri and collaborators that one needs to ...

**4**

votes

**0**answers

236 views

### Reference request: virtual fundamental class of moduli of stable maps

Let $f:U\longrightarrow \overline{M}_{0,n}(\mathbb{P}^m,d)$ is the universal family with morphism $\pi:U\longrightarrow\mathbb{P}^m$ and let $X\subset\mathbb{P}^m$ be a hypersurface defined by a ...

**4**

votes

**0**answers

256 views

### Applications of moduli of curves theory

Are there some applications of moduli of curves theory? I was wondering if moduli of curves theory is used (or could be used) for doing research in applied mathematics.
I am doing my PhD in algebraic ...

**4**

votes

**0**answers

237 views

### Picard group of $M_{g,n}$

It is well known that the Picard group of $M_g$ is $\mathbb{Z}$ generated by the Hodge bundle. What is the Picard group of $M_{g,n}$, the moduli space of curves with marked points?

**4**

votes

**0**answers

182 views

### Properties of the Zariski-Riemann topology on the set of valuations

One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...

**3**

votes

**0**answers

169 views

### Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, ...

**3**

votes

**0**answers

133 views

### Properties of finite quotients of quasi-projective varieties

Let $G$ be a finite group acting on a (smooth) quasi-projective variety over $\mathbb C$.
One can consider the stacky quotient $[X/G]$ or the "classical" quotient $X/G$. In general, $[X/G]$ is not a ...

**3**

votes

**0**answers

88 views

### Examples of subspaces singled out by modular forms

I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature.
To be more clear let me explain the example I have in mind.
Let ...

**3**

votes

**0**answers

349 views

### A quotient stack question

Let $X$ be a proper Deligne-Mumford stack, whose normalization, $X'$, is a global quotient stack (that is, a stack of the form [W/GL_n],where W is an algebraic space) with a projective scheme as a ...

**3**

votes

**0**answers

110 views

### state of the art for kodaira dimension of $\overline{\mathcal{M}}_{g,n}$

What is the state of the art about the kodaira dimension (and rationality, unirationality, etc.) of the moduli spaces of $n$-pointed curves of genus $g$? When it is known and when not? It would be ...

**3**

votes

**0**answers

224 views

### Stack of vector bundles (on a curve) over a strictly semi-stable point of the moduli space

Consider the stack $Bun_{r,d}^{ss}$ of rank $r$ semi-stable vector bundle of degree $d$ over a fixed curve. There exists also a coarse moduli space $M$ built via GIT. Over the stable locus of $M$ it ...

**3**

votes

**0**answers

132 views

### VHS for universal family of false elliptic curves

If $f: X \rightarrow C$ is the universal family of false elliptic curves (i.e. abelian surfaces $A$ such that $End(A) \otimes \mathbb{Q}$ is a totally indefinite quaternion algebra over $\mathbb{Q}$), ...

**3**

votes

**0**answers

180 views

### Does the Albanese map satisfy Torelli's theorem

Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A_g$ the moduli space ...

**3**

votes

**0**answers

257 views

### Every curve is a Hurwitz space in infinitely many ways

Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space.
A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the ...

**3**

votes

**0**answers

211 views

### ubiquitous modulicity?

On the one hand, as mentioned here, basically "everything" in algebraic geometry could be seen in the context of "moduli problems" - on the other hand, Grothendieck's few remarks on a possible "tame ...

**3**

votes

**0**answers

202 views

### How can one check that two line bundles on $\overline{M}_{0,n}$ coincide?

Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of ...