**0**

votes

**2**answers

315 views

### The non-existence of the fine moduli scheme of vector bundles. Why?

The reference I am using is this one. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a projective curve. Let $S$ ...

**1**

vote

**0**answers

93 views

### On dimension of the moduli space of abelian differentials on Riemann surfaces

I fear I'm missing something important here, so forgive me if my question is stupid.
Consider $\mathcal{M}_g$ the moduli space of Riemann surfaces of genus $g>2$ and $\mathcal{H}_g$ the moduli ...

**1**

vote

**0**answers

106 views

### Is there a reference for boundedness of smooth canonically polarized varieties over Z (No…)

In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack $\mathcal M_P$ of smooth canonically polarized varieties over Spec $\mathbb Z$ with ...

**4**

votes

**0**answers

102 views

### Lifting morphisms of $p$-divisible groups using Grothendieck-Messing theory

During my reading of Peter Scholze and Jared Weinstein's paper ``Moduli of $p$-divisible groups'' I found this assertion in the proof of Proposition 6.1.3. Consider the following situation. Let $k$ be ...

**8**

votes

**0**answers

299 views

### How do I complete the sketch of proof in 'FGA explained', 5.5.8?

It seems to me that the one part that is difficult to transfer to general coherent sheaves is given only one sentence: "Because we have such a common $m$, we get as before an injective morphism from ...

**5**

votes

**1**answer

152 views

### Regular minimal model of $X_0(p^2)$

Consider the compactified modular curve $X_0(p^2)$ and the corresponding algebraic curve over $\mathbb{Q}$. My questions are the following:
Where do the cusps of $X_0(p^2)_{\mathbb{Q}}$ live? That ...

**2**

votes

**0**answers

131 views

### Are these moduli problems of curves “well-behaved”?

Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.
Let $H_{X,d}$ be the Hilbert scheme ...

**2**

votes

**0**answers

104 views

### Dessin d'enfant and moduli space of bordered/punctured hyperbolic Riemann surfaces

Belyi's theorem states that if a Riemann surface could be defined as an algebraic curve over an algebraic number field, then this Riemann surface could be described by a Dessin d'enfant. I have two ...

**3**

votes

**0**answers

59 views

### Psi-classes on moduli spaces of weighted curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...

**2**

votes

**0**answers

120 views

### Abstract VFC vs. what people actually use for Quintic 3-fold

Moduli space of genus $0$ degree $d$ maps in a quintic Calabi-Yau threefold $X$, written as $\overline{\mathcal{M}}_{0,0}(X,[d])$, can be embedded in the corresponding moduli space of $\mathbb{P}^4$, ...

**3**

votes

**0**answers

254 views

### Restriction of a global moduli functor that admits a coarse moduli space

Let $F:(Sch/k)^{o}\to Sets$ be a functor, where $Sch/k$ is the category of schemes over a field $k$. Suppose that $F$ admits a coarse moduli space, let it be $M$. Consider a $k$-point $x\in M$ (which ...

**3**

votes

**0**answers

96 views

### Some questions on Kontsevich's moduli space

Motivation: Work of Eisenbud, Harris, and Mumford shows that
$\mathcal M_g$ is of general type when $g≥24$. Moreover, by Logan's function $f(g)$ , $\overline {\mathcal M_{g,n}}$ is of general type for ...

**3**

votes

**0**answers

77 views

### Effective divisor in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-classes

Does anybody knows an effective class in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-coefficients? The standard references; Logan, Farkas or Brill-Noether divisors have all non-negative ...

**2**

votes

**0**answers

97 views

### Dimension of a sheaf cohomology group on a genus 1 curve

Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...

**1**

vote

**0**answers

120 views

### A question about canonical bundle of moduli space of Kahler Einstein metrics

Let $\mathcal M$ be a moduli space of Kahler-Einstein metrics with
negative Ricci curvatures on pairs $(X,D)$. Is the canonical bundle of
$\mathcal M$ nef?
Motivation: If we know the nefness of $...

**4**

votes

**1**answer

176 views

### Special fibre of the modular curve $X(N)$

Let $N$ be an integer $\geq 3$ and $X(N)\rightarrow \mathrm{Spec } \mathbb{Z}[1/N]$ is the projective smooth modular curve defined in Deligne-Rappoport. Is there an exemple of $N$ for which the ...

**8**

votes

**2**answers

481 views

### Moduli space of (all) vector bundles on $\mathbb{P}^1$

It is well known that, by a theorem of Grothendieck, every vector bundle (always assumed coherent in this question; and everything is over the complex numbers) splits as a direct sum of line bundles.
...

**4**

votes

**1**answer

189 views

### Deformation long exact sequence of GW theory in the analytical setting

Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume $C=(\...

**1**

vote

**0**answers

100 views

### Components of Kontsevich moduli space of stable maps and reducible curves

Let $X\subset \mathbb P^n$ be a smooth projective variety which is Fano and $M_{0,0}(X,e)$ the (projective) Kontsevich moduli space of rational curves of degree $e>1$. Is it possible (or are there ...

**1**

vote

**1**answer

91 views

### Components of Kontsevich moduli space of stable maps and multiple covers

Let $X\subset \mathbb P^n$ be a smooth projective variety over $\mathbb C$ which is Fano and $M_{0,0}(X,e)$ the (projective) Kontsevich moduli space of rational cuves of degree $e>1$. Is it ...

**8**

votes

**1**answer

307 views

### Topology on the space of constructible sheaves

Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...

**3**

votes

**0**answers

93 views

### Moduli space of null Sasaki $η$-Einstein structures for higher dimensions(Calabi-Yau structures in Sasakian setting)

The moduli space of null Sasaki $η$-Einstein structures for simply connected compact 5-dimensional manifold $M$ is determined by the following quadric
$$\{[\alpha]\in H^2(M,\mathbb C) \; \text{such ...

**4**

votes

**1**answer

187 views

### Essential dimension and the moduli space of abelian varieties

The following problem is listed here: http://www-personal.umich.edu/~erman/Papers/Questions2.pdf and attributed to Vistoli:
Let $\mathcal A_g$ denote the moduli stack of principally polarized abelian ...

**1**

vote

**0**answers

66 views

### Complex Structure Moduli of Elliptic Fibrations

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.
I know how it is done for the generic Weierstrass ...

**2**

votes

**0**answers

64 views

### Symplectic structure moduli of simple bundles on hyper-Kaehler manifolds

Let $S$ be a K3 or Abelian surface and let $M_{S}$ be a moduli of stable bundles on $S$. Then, Mukai proves that there $M_{S}^{H}$ has a symplectic structure. Indeed, let $\mathcal{F}$ be the ...

**9**

votes

**4**answers

329 views

### Moduli spaces in applied mathematics and condensed matter physics?

In this MO question it is stated that there is a relation between some aspects of condensed matter physics (namely the fractional quantum Hall effect) and the algebraic geometry of Hilbert schemes.
...

**10**

votes

**1**answer

258 views

### Counting isomorphism classes in open subsets of Bun_G

Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$.
The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each ...

**4**

votes

**0**answers

105 views

### Moduli space of complex Tori [duplicate]

Is there any explicit computation for the Weil-Petersson metric on moduli space of Tori of complex dimension n?

**3**

votes

**0**answers

268 views

### Moduli space of log Calabi-Yau varieties exists?

Let $\mathcal M^{(X,D)}$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor with conic singularities on Kaehler variety $X$. I am looking for a proof that such ...

**2**

votes

**1**answer

111 views

### What finite groups are stabilizers in Kirwan's desingularization construction?

Assume $X$ is a smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$ and let $M$ be the (singular) moduli space of semistable rank two vector bundles with trivial determinant on $X$. Then ...

**7**

votes

**1**answer

275 views

### When is a general sheaf (on the projective plane) globally generated?

Let $v$ be a chern character on $\mathbb P^2$ so that the moduli of sheaves of chern character $v$ is non-empty of the expected dimension. When is it true that the general sheaf in moduli is globally ...

**5**

votes

**0**answers

99 views

### Flatness of Chern classes for flat family of sheaves

Let $Q$ be a quasi-projective $k$-scheme (not necessarily smooth), $X$ a smooth projective $k$-variety and $\mathcal E$ a family of (torsion free) sheaves on $X$ parametrized by $Q$. Suppose that $\...

**0**

votes

**0**answers

101 views

### Chern classes of a family and Chern classes of a member

Let $X$ be a smooth projective variety over an algebraically closed field $k$ and $\mathcal E$ a family on torsion free coherent sheaves on $X$ parametrized by a smooth curve (over $k$) i.e. a ...

**1**

vote

**0**answers

44 views

### Understanding the Exp map from a moduli of smooth curves

The setup:
Suppose I'm given a family of class $C^k$ curves $\{f_{z}\}_{z\in Z}$ with values in $\mathbb{R}$ parameter by a finite number of parameters $Z\subseteq \mathbb{R}^m$.
Let $\mathscr{M}$ be ...

**2**

votes

**1**answer

209 views

### Moduli of stable bundles - analytic approach

Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.
At that point, one states ...

**4**

votes

**0**answers

169 views

### Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference
in the literature...
Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...

**3**

votes

**0**answers

283 views

### Lifting a real quadratic twist of an Elliptic Curve to the modular curve

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
...

**5**

votes

**1**answer

342 views

### Relations between some works by Deligne-Mostow and Thurston

Happy new year 2016!
A coworker and I are interested in the relations between the works of Deligne and Mostow ([DM] and [M]) on the monodromy of Appell-Lauricella hypergeometric functions (Publ. ...

**1**

vote

**0**answers

220 views

### Weil-Petersson metric is quasi isometric with which model?

Let $\mathcal M_g$ be the moduli space of curves of genus $g$. If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow
$X^{reg}$ with a complete Kahler metric which has a ...

**1**

vote

**0**answers

113 views

### Push-forwards of some codimension 2 classes from the universal curve to $\overline{\mathcal{M}}_{g,n}$

Let $\pi:\overline{\mathcal{M}}_{g,n+1}\to \overline{\mathcal{M}}_{g,n}$ be the map that forgets the last marked point and $\omega_\pi$ the relative cotangent line bundle (Here we are identifying $\...

**17**

votes

**2**answers

901 views

### elliptic curves and group cohomology

Recently, I've been trying to understand Jacob Lurie's 2-equivariant elliptic cohomology a bit better than I had in the past.
From what I can tell, the fragment of the story that only deals with ...

**2**

votes

**0**answers

179 views

### A question about Kobayashi hyperbolic variety

Let $X$ be a projective variety with snc divisor $D$ with $K_X+D<0$, which pair $(X,D)$ is log k-stable
.Take $\mathcal M$ be the moduli space of log Fano
pairs. Then the regular part $\mathcal ...

**2**

votes

**0**answers

175 views

### Integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of varieties of general type is a rational numbers?

It is known that the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of Calabi-Yau varieties is a rational numbers
My question is about the moduli space of ...

**3**

votes

**2**answers

201 views

### How do we get the quotient $Ext^1(N,M)/Hom(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of torsion free coherent sheaves on a surface. Here $M$ is a line bundle, $E$ a vector bundle of ...

**8**

votes

**1**answer

242 views

### Intuitive reasons for the existence of modular parametrizations

Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object ...

**4**

votes

**1**answer

144 views

### A moduli problem inspired by Stein factorization

Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ ...

**12**

votes

**1**answer

805 views

### What's wrong with my understanding of the scheme $\text{Isom}(E_\lambda, E_{\lambda'})$?

Let $\mathcal{M}_{1,1}$ be the moduli stack of elliptic curves (over the complex numbers). Define
$$\begin{eqnarray*} X &:=& \Bbb{A}^1_{\lambda} - \{0,1\}\\
X' &=& \Bbb{A}^1_{\lambda'} ...

**11**

votes

**0**answers

200 views

### Are Hecke eigenvalues on the cohomology of the Newton polygon strata automorphic?

Fix a genus $g$, a prime $p$, and a Newton polygon $\Delta$ of an abelian variety of genus $g$.
Let $\mathcal A_{g, \overline{\mathbb F}_p, \Delta}$ be the moduli stack of abelian varieties of genus $...

**4**

votes

**2**answers

133 views

### Explicit constant terms of volumes of moduli spaces

In Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Mirzakhani gave a recursive formula for WP volumes of moduli spaces $\mathcal{M}_{g,n}(L)$ of bordered ...

**8**

votes

**1**answer

329 views

### Automorphisms of curves in positive characteristic

It is well known that over an algebraically closed field of characteristic zero a general curve (for an open subset of $M_g$) of genus $g\geq 3$ is automorphism-free.
Is this result still true over ...