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24
votes
0answers
477 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 ...
15
votes
0answers
355 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). ...
11
votes
0answers
188 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
0answers
655 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
0answers
975 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
0answers
359 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 ...
10
votes
0answers
326 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
0answers
684 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
0answers
310 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 ...
9
votes
0answers
327 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
0answers
326 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
0answers
254 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 ...
8
votes
0answers
274 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
0answers
501 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 ...
6
votes
0answers
150 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
0answers
557 views

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

Is anyone else working through this paper : http://arxiv.org/abs/0810.5645 ? I am trying to verifying example 6.2 (m=2 for simplicity) using only the definitions, namely: J^{2\alpha}(\tau) = -1/4 ...
5
votes
0answers
123 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
0answers
219 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 ...
4
votes
0answers
118 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
0answers
92 views

On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces

There are many results on the cohomology of the Hilbert scheme of points of a surface. Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also there are results ...
4
votes
0answers
105 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
0answers
127 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
0answers
231 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
0answers
56 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
0answers
282 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
0answers
174 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
0answers
164 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 ...
4
votes
0answers
225 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
0answers
251 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
0answers
230 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
0answers
179 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. ...
4
votes
0answers
574 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 ...
3
votes
0answers
118 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. ...
3
votes
0answers
332 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
0answers
132 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 ...
3
votes
0answers
104 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
0answers
191 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
0answers
123 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
0answers
159 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
0answers
238 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
0answers
205 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
0answers
197 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 ...
3
votes
0answers
545 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 ...
2
votes
0answers
105 views

Symplectic form on moduli space of connections

Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it. Is there any known construction of a bundle with a ...
2
votes
0answers
79 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 ...
2
votes
0answers
154 views

Can Kuranishi families glue together to give a moduli space?

In his article "Moduli spaces of vector bundles on K3 surfaces and symplectic manifolds", Mukai constructs moduli of simple sheaves on a K3 surface through a theorem that asserts the existence of a ...
2
votes
0answers
84 views

Moduli space of sheaves on a ribbon

In the paper "A non-linear deformation of the Hitchin dinamycal system", Donagi-Ein-Lazarsfeld describe the irreducible components of the moduli space $\mathcal M_R$ of stable sheaves of numerical ...
2
votes
0answers
49 views

Characterisation of convergence in Deligne-Mumford compactifiaction

1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex ...
2
votes
0answers
208 views

Beautiful curves in Gromov-Witten theory, and in Donaldson-Thomas theory

Let $X$ be a smooth complex projective threefold, and $\beta\in H_2(X,\mathbb Z)$ a curve class. In the Kontsevich's moduli space of stable maps, $\overline M_g(X;\beta)$, a general point $[f:C\to X]$ ...
2
votes
0answers
206 views

Cech cohomology.

Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with ...