**8**

votes

**1**answer

416 views

### Mordell-Weil group of the universal abelian scheme

Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
...

**20**

votes

**6**answers

2k views

### Does the moduli space of smooth curves of genus g contain an elliptic curve

Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete.
Does $M_g$ contain an elliptic curve?
The answer ...

**6**

votes

**1**answer

390 views

### Deformations of smooth projective hypersurfaces and the Jacobian ring

It is a well-known result of Griffiths that the pieces of Hodge filtration of a smooth hypersurface $X:= (f=0)$ of degree $d$ in $\mathbb{P}^{n}$ are isomorphic to graded pieces of the Jacobian ring ...

**5**

votes

**1**answer

356 views

### Is $M_{1,n}$ affine?

A famous conjecture of Looijenga states that the moduli space of curves $M_{g,n}$ is the union of $g- \delta_{0,n}+ \delta_{0,g}$ open affine subsets, where $g,n$ are non-negative integers satisfying ...

**5**

votes

**1**answer

457 views

### Some help in digesting a paragraph in the introduction of Deligne/Rapoport's “Les Schemas de Modules de Courbes Elliptique”

http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf
On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as ...

**1**

vote

**1**answer

264 views

### Fine/Coarse moduli spaces and extensions of fields.

Let $K/k$ be an arbitrary field extension and $X$, $Y$ varieties over $k$ (lets assume projective and perhaps smooth to avoid technicalities). There is a fine moduli space of morphisms between $X$ and ...

**1**

vote

**0**answers

151 views

### Irreducibility of monodromy of eigenspaces of families of cyclic coverings

In the article "La conjecture de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in ...

**3**

votes

**2**answers

407 views

### Moduli Space of Abelian Varieties with a N-torsion point

Does there exists (as scheme, or as some sort of stack) the moduli space of principally polarized Abelian Varieties together with a point of order $N$, for $N>1$ an integer?
In the case of ...

**9**

votes

**1**answer

338 views

### Koszulness of the cohomology ring of moduli of stable genus zero curves

Let $n \geq 3$. The ring $H^\bullet(\overline{M}_{0,n},\mathbf Q)$ was determined by Sean Keel. It is generated by the cohomology classes of boundary divisors $D_{A,B}$ corresponding to partitions $A ...

**27**

votes

**1**answer

834 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

228 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

166 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

244 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

347 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

212 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

305 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

458 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

253 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

475 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

235 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

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 $g$ ...

**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

372 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

363 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

347 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

345 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

272 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

209 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

281 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

658 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

288 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

461 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

398 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

188 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

338 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 ...