**27**

votes

**1**answer

827 views

### Is $M_g$ finitely covered by a scheme over the integers?

This question was prompted by my almost-answer to the question Does smooth and proper over $\mathbb Z$ imply rational? , but I never got around to asking this until now.
It is well known that $M_g$, ...

**2**

votes

**1**answer

227 views

### Deformations of pointed stable maps with “curve held rigid” or “preserving the dual graph”

I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order ...

**1**

vote

**0**answers

165 views

### Smooth curve in the Hilbert flag scheme

Let $d$ be an integer greater than $0$. Let $P_2$ be the Hilbert polynomial of a degree $d$ surface in $\mathbb{P}^3$. Recall, the Hilbert flag scheme $\mathrm{Hilb}_{P_1,P_2}$ parametrizes curves $C$ ...

**3**

votes

**1**answer

238 views

### Representability of Hom-sheaves of various moduli spaces

(May be a poor title, happy to update)
Recall that for a stack $\mathcal{X} \to Sch$ on schemes (e.g. fppf site) and a pair of morphisms $x,y\colon U\to \mathcal{X}$ ($U$ a representable stack) there ...

**9**

votes

**0**answers

344 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 ...

**0**

votes

**0**answers

211 views

### Intersections with divisors on moduli of curves

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of stable curves of genus $g$ with $n$ points.
Consider
$0 \neq \gamma \in Pic(\overline{\mathcal{M}}_{g,n})$
the first Chern class of a ...

**5**

votes

**1**answer

303 views

### Modular curve parametrizing two cyclic subgroups of an elliptic curve

The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing ...

**3**

votes

**2**answers

453 views

### Degeneration of projective curves

Given a projective curve $C$, is it possible that $C$ can degenerate into union of lines i.e., does there exist a family of curves $\pi:\mathcal{C} \to B$ such that $\pi^{-1}(0)=C$ and there exists $a ...

**1**

vote

**0**answers

147 views

### Hurwitz Spaces and Rauch Variational Formulas

I have read in some papers about Rauch-type variational formulas on Hurwitz spaces, and I would like to know what exactly is the theory behind them.
A Hurwitz Space $H_g^d$ is the space of coverings ...

**3**

votes

**0**answers

251 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

**1**answer

423 views

### Wanted: differential coming from higher genus surface in Heegaard Floer Homology

I am interested in studying moduli of complex surfaces which arise in computing the differential on the Heegaard Floer Homology chain complex. In particular, I am interested in the generic case, when ...

**2**

votes

**0**answers

168 views

### Level n-structure as defined by Mumford in GIT

In Mumford's GIT, the definition of level $n$ structure ($n \geq 2)$ is $2g$ sections $\{\sigma_1, \dots, \sigma_{2g}\} : S \rightarrow A$ such that two conditions hold: (i) For geometric points the ...

**4**

votes

**0**answers

234 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?

**3**

votes

**1**answer

285 views

### How big is the locus of Galois covers in the moduli space of curves

Let's consider the moduli space $M_g$ of curves of genus $g$ over $\mathbf{C}$.
Not every curve of genus $g$ is a Galois cover (of the projective line) if $g\geq 3$.
How big is the locus of Galois ...

**32**

votes

**2**answers

2k views

### Meaning/Origin of Seiberg-Witten Equations/Invariants

Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take an ...

**6**

votes

**0**answers

163 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 $g$ ...

**2**

votes

**1**answer

200 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

**1**answer

371 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

**1**answer

807 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

**0**answers

362 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

**0**answers

345 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

**0**answers

341 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

**2**answers

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

**0**answers

161 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

**1**answer

270 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

**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

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

**0**answers

279 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

**1**answer

648 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

**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 ...

**5**

votes

**1**answer

149 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

**4**answers

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

**0**answers

210 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

**1**answer

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

**1**answer

287 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 ...

**4**

votes

**1**answer

456 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

**2**answers

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

**1**answer

525 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

**3**answers

396 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

**2**answers

187 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

**1**answer

334 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

**1**answer

517 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

**0**answers

696 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

**1**answer

404 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

**1**answer

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 ...

**30**

votes

**4**answers

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

**1**answer

584 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

**3**answers

462 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

**1**answer

419 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

**2**answers

434 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 ...