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 ...

learn more… | top users | synonyms

2
votes
1answer
196 views

$\psi$ class in $\overline{M}_{0,n}$

Basic question, but I found no reference. Is the $\psi$ class the only one which is not a boundary class in the PIcard group of the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$? Or can it ...
2
votes
1answer
359 views

fano moduli varieties of vector bundles

Let $M$ be a fine moduli space of vector bundles on curve which is an algebraic variety as well. The first example of such an object that I have in mind is rank 2, deg 1 VB on a genus 2 curve. This is ...
1
vote
1answer
802 views

Penner's formula for volume of the Moduli Space

In his paper "Weil-Petersson Volumes" Penner gives the following formula for the integral of a top-dimensional cohomology class $\omega$ on the moduli space $\mathcal M_g^s$ of $s$-punctured riemann ...
0
votes
0answers
353 views

modular forms of Gamma^0(N) with some Dirichlet character

I am just a beginner in modular forms.. It seems for me that lots of the work has been done for the cases of spaces of modular forms for $\Gamma_0(N)$ or $\Gamma_1(N)$ with some Dirichlet character ...
10
votes
0answers
336 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 $\pi_1(\Sigma)$. ...
9
votes
0answers
336 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 ...
25
votes
2answers
2k views

Is the moduli space of curves defined over the field with one element?

There are various frameworks around which enlarge the category of rings to include more exotic objects such as the 'field with one element,' $\mathbb{F}_1$. While these frameworks differ in their ...
0
votes
0answers
160 views

What means “extended concepts of symmetry”?

Where could one find a short description oft: "two mathematical extensions of the symmetry - to moduli spaces of sheaves and to derived categories", found here? Happen there interesting things like ...
3
votes
1answer
260 views

Applications of the boundedness of birational automorphisms

Recently the paper http://de.arxiv.org/PS_cache/arxiv/pdf/1011/1011.1464v1.pdf by Hacon, McKernan and Xu appeared on the arXiv. There the authors prove that the number of birational automorphisms of a ...
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. ...
3
votes
0answers
208 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 ...
2
votes
0answers
275 views

Moduli space of points of fixed order N on elliptic curves

Let us consider the moduli space $Y_1(N)$, parametrizing an elliptic curve, together with a choice of a point of N-torsion on it. Over the complex numbers, it is easy to see that this moduli space is ...
7
votes
1answer
607 views

Geometric interpretation of Simpson's correspondence

What is the exact geometric meaning of the Simpson's correspondence between Higgs bundles and local systems ? I know that it should have a rich geometric content but don't know an explicit geometric ...
3
votes
0answers
201 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 ...
5
votes
1answer
145 views

Stable reduction for maps

I would like to explicitly compute the limit of a family of stable maps in $\overline{M}_{0,n}(\mathbb{P}^r,d)$. I know in principle how this works for families of curves without maps as I found a ...
9
votes
4answers
1k views

Moduli space of genus 2 curves

Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied?
2
votes
0answers
208 views

Is there an algebraic analogue of the degeneration of riemann surfaces in M_g

Degeneration of certain functions such as theta functions or Green's functions in the moduli space $\overline{\mathcal{M}_g}$ of stable curves of genus $g$ has been studied quite alot. The idea is to ...
3
votes
1answer
237 views

lifting the isomorphisms between abelian schemes over PD thickenings

Assume that X and Y are abelian schemes (or even abelian varieties) over a base T. If $\ S --> T$ is a PD nilpotent thickening (i.e. the ideal of $\ S$ in $ \ T$ is a nilpotent divided power ...
1
vote
1answer
281 views

The locus of cyclic covers in the moduli space of curves

Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. Let $Z$ be the closure in $\mathcal{M}_g$ of the set of smooth curves of genus $g$ which are a cyclic cover of the projective ...
3
votes
1answer
429 views

Virtual dimension of moduli of stable maps

I apologize if this question was already asked somwhere else on this website. Let us consider $f:C \to X$ a stable map. This is a point in the moduli space of stable maps. It seems intuitively to me ...
2
votes
2answers
261 views

Reference request: parametrizing covers of the projective line

Hurwitz spaces (or Hurwitz schemes) parametrize covers of the projective line. One can do this in many ways. For example, one could fix the number $r$ of branch points, the degree $n$ of the cover ...
3
votes
1answer
520 views

moduli of vector bundles on a surface

Let $S$ be a smooth projective surface with an ample divisor $X\subset S$. Consider the moduli stack of vector bundles $F$ on $S$ such that 1) $c_1(F)=0$ 2) $c_2(F)=n$ 3) The restriction of $F$ to ...
8
votes
3answers
392 views

Applications for knowing the singularities parametrized by the boundary of a moduli space

Given a moduli space $M$ of some smooth algebraic geometric object such as curves, surfaces, etc. Let $\overline{M}$ be a compactification of $M$. Then, $\overline{M}\setminus M$ introduces ...
2
votes
2answers
184 views

Is restricting the support of an Artinian sheaf a closed condition?

Given a projective surface $S$, and a smooth projective curve $C\subset S$ over $\mathbb{C}$. Furthermore we have a locally free sheaf $E$ of rank $r$ on $S$. Then for any $l\geq 1$, the projective ...
8
votes
1answer
324 views

Given a curve, under which condition is the set of gonal morphisms finite

Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that ...
13
votes
1answer
512 views

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 ...
10
votes
0answers
690 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 ...
7
votes
1answer
393 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
votes
1answer
267 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 ...
29
votes
4answers
2k views

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 ...
17
votes
1answer
564 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 ...
7
votes
3answers
457 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
votes
1answer
418 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 ...
4
votes
2answers
428 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
votes
3answers
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 ...
8
votes
1answer
573 views

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, ...
12
votes
1answer
1k views

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 ...
10
votes
2answers
463 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
votes
1answer
291 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 ...
8
votes
2answers
457 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 ...
5
votes
2answers
419 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:           ...
12
votes
3answers
1k views

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 ...
9
votes
3answers
1k 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 ...
2
votes
1answer
407 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 ...
3
votes
0answers
559 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 ...
24
votes
0answers
486 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 ...
1
vote
2answers
443 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 $Quot(E,l)$ ...
2
votes
0answers
213 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 ...
8
votes
1answer
310 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 ...
3
votes
1answer
246 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, ...