**4**

votes

**1**answer

97 views

### BCOV's holomorphic anomaly equation at genus one

BCOV in their famous paper (http://arxiv.org/abs/hep-th/9309140) state the genus one holomorphic anomaly equation (on page 53) to be $$\partial_i \partial_{\bar{j}} F_{1} = ...

**6**

votes

**1**answer

197 views

### Do general sheaves on P^2 have cohomology governed by their Euler characteristic?

Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$.
If $\xi$ has Euler characteristic 0, then apparently there is ...

**6**

votes

**0**answers

217 views

### Is the stack of varieties with a big line bundle algebraic

In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...

**8**

votes

**0**answers

267 views

### Coherent cohomology of the moduli space of curves

Is $H^i\left(\overline{\mathcal M}_g, \mathcal O_{\overline{\mathcal M}_g}\right)$ nontrivial for any $i>0$ and any $g$?
I was not able to find literature on this after searching for a bit, ...

**0**

votes

**0**answers

100 views

### Hilbert scheme of relative subschemes of lenght 2

Let $\mathfrak X \rightarrow S$ a smooth projective family over the spectrum of a dvr. We know that $(\mathfrak X _{\eta_R})^{[2]}$ and $(\mathfrak X _{p})^{[2]}$ are smooth, where $p$ is the closed ...

**7**

votes

**1**answer

440 views

### Do modular forms show up in the cohomology of moduli spaces of unmarked curves?

Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + ...

**0**

votes

**0**answers

150 views

### Is the complex structure on a del-Pezzo surface a regular complex structure?

Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost
complex structure $J$ is said to be ...

**1**

vote

**0**answers

73 views

### local universal sheaf (moduli of stable sheaves)

I do not know much about moduli of sheaves and I wanted to shows that for a smooth (projective) family over a discrete valuation ring of mixed characteristics (relative dimension 3), the locally free ...

**0**

votes

**1**answer

135 views

### Complex plane mod lattice to elliptic curve correspondence generalization

If we observe the correspondence
$$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g_{2}(\Lambda)}{4}X - \frac{g_{3}(\Lambda)}{4},$$
we see the relationship between weight 4 and weight 6 ...

**4**

votes

**1**answer

163 views

### Non canonical singularities of moduli spaces of curves

Is it true that for any $g\geq 1$ and $n$ such that $\overline{M}_{g,n}$ has dimension at least two the locus in $\overline{M}_{g,n}$ parametrizing reducible curves which are union of an elliptic ...

**0**

votes

**1**answer

159 views

### When is the normal neighbourhood of the boundary of the moduli space of cuvres parametrized by exactly one branch?

Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z}) $
a fixed homology class that is $\textit{decomposable}$. Let
$$ \overline{\mathcal{M}}_{0,n}(X, \beta) $$
denote the stable ...

**0**

votes

**0**answers

174 views

### How does one define Moduli spaces in Symplectic Geometry and naively interpret higher genus GW Invariants?

This is a very basic question about the definition of Moduli space of maps.
My reason for asking this question is because I haven't actually seen this
definition explicitly given anywhere, and hence ...

**1**

vote

**0**answers

137 views

### Moduli space of line-bundle holomorphic structures and sections over a Kahler-Hodge manifold

Let $(M,\omega)$ be a Kahler manifold with Kahler integral two-form $\omega$ and let $(L,h)$ be a rank-one complex vector bundle over $M$ equipped with a fixed hermitian metric $h$. I am interested in ...

**2**

votes

**1**answer

227 views

### Rational curves in projective spaces

Let $X\subset(\mathbb{P}^{N})^n$ be the variety defined as follows: $(p_1,...,p_n)\in (\mathbb{P}^{N})^n$ such that there exists a rational curve $C$ of degree $d$ with $p_1,...,p_n\in C$.
Is there a ...

**1**

vote

**1**answer

333 views

### Soft Question: Relationships Between Moduli Space and Objects They Parametrize

Apologies in advance if this question is not suitable for MO. My friend and I were wondering recently what, if any, are the relationships between the geometric properties of a moduli space and the ...

**1**

vote

**0**answers

89 views

### On the compactification of moduli space of vector bundles

Let $X$ be an irreducible, nodal curve over an algebraically closed field of genus at least $2$. Denote by $U(r,d)$ (resp. $U^0(r,d)$) the moduli space of torsion-free (resp. locally free) sheaves of ...

**4**

votes

**2**answers

213 views

### actual dimension of concrete moduli space of holomorphic curves vs its virtual dimension

I am looking at exercise 6.3.3 in Mcduff's and Salamon's book J-holomorphic curves and Symplectic topology, which basically gives an example of a moduli space whose actually dimension is greater than ...

**2**

votes

**1**answer

297 views

### An application of the Base Change Theorem to the moduli space of sheaves

Let $S$ be an abelian or K3 surface, $H$ an ample class on $S$, $v\in H^{ev}(S,\mathbb{Z})$ and $M$ the moduli space of $H$-stable sheaves on $S$ with invariants fixed by $v$. Let $\mathcal{E}$ be a ...

**0**

votes

**0**answers

76 views

### Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?

Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...

**4**

votes

**1**answer

121 views

### Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question.
Is ...

**1**

vote

**0**answers

106 views

### Is it possible to find an explicit definition of the “universal” (co)tangent bundle?

Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$
maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point
$y \in \mathbb{P^1} $ ...

**3**

votes

**0**answers

193 views

### Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, ...

**29**

votes

**2**answers

820 views

### Volume of the unitary group

I saw a very remarkable asymptotic formula (or a conjecture?) for the volume of of the unitary group $ U(n)$ which is the following:
$$\log[\mathrm{Volume}(U(n))] \sim_{n\rightarrow \infty} ...

**7**

votes

**1**answer

302 views

### Singularities of the moduli stack of Calabi-Yau threefolds

Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack?
In many cases I ...

**0**

votes

**1**answer

134 views

### Reflexive sheaves on stable curves

Let $C$ be a stable curve over an algebraically closed field of positive characteristic and $\mathcal{F}$ be a reflexive sheaf on $C$. Is $\mathcal{F}$ locally free?
EDIT Is the projective dimension ...

**6**

votes

**0**answers

181 views

### Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a ...

**4**

votes

**2**answers

327 views

### Universal curve of stacks of stable curve

Let $\overline{M}_{g,A}$ the moduli stack of pointed genus $g$ stable curves with weights $A = (a_1,...,a_n)$ introduced in
Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. ...

**0**

votes

**0**answers

120 views

### Moduli space of holomorphic sections

Let $(L,M,\omega,\nabla)$ be an holomorphic line bundle over a Kahler manifold $(M,\omega)$ equipped with the Chern connection $\nabla$. Let $\Gamma(L)$ denote the space of holomorphic sections of ...

**0**

votes

**1**answer

173 views

### Need help on explanations of moduli space (of genus 1 curves) [closed]

Sorry about this question which is not on a research level. But I am very confused about this "first" example of coarse moduli space of genus 1 curves.
The moduli space I talk about here is the ...

**1**

vote

**1**answer

220 views

### Moduli space of flat connections over a torus

Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...

**5**

votes

**1**answer

213 views

### Evaluation maps for moduli of stable maps

Let $\overline{M}_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$.
Consider the product of the evaluation ...

**4**

votes

**0**answers

107 views

### $\mathcal{M}_{g,n}$ a scheme for $n \gg 0$? [duplicate]

I think that for $n \geq 3$, the Deligne-Mumford moduli stack $\mathcal{M}_{0,n}$ is a scheme. Is it more generally true that for every $g$, the Deligne-Mumford moduli stack $\mathcal{M}_{g,n}$ is a ...

**9**

votes

**1**answer

514 views

### A question about Mirzakhani et. al.'s algebraicity theorem

While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...

**1**

vote

**0**answers

75 views

### miniversality vs versality

Consider a moduli problem $\mathcal{M}$. Assume, at each point $x$, the associated deformation problem $\mathcal{M}_x$ has a tangent-obstruction theory. It follows that $\mathcal{M}_x$ has a hull ...

**31**

votes

**5**answers

3k views

### Maryam Mirzakhani's works

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...

**5**

votes

**0**answers

155 views

### Is the moduli space of curves arising from wild ramification smooth?

Fix a natural number $g$, a prime $p$, and a $p$-group $P$.
Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...

**3**

votes

**0**answers

87 views

### degenerate points in the moduli space of flat principal $G$-bundle with respect to a linear representation on a complex

Let $(A^\bullet,\partial)$ be a complexe of $\mathbb{C}$-vector spaces. We suppose that this complex is of finite length, and all $A^\bullet$ are finite dimensional. Let $H^\bullet$ be the cohomology ...

**1**

vote

**1**answer

240 views

### Proof of “generic curve of genus at least 2 has no nontrivial maps to a positive genus curve”

I searched for it for a long time, but it seems that everybody is taking this for granted and does not bother to point out a proof. Would it be possible that someone points me to a proof or makes me ...

**2**

votes

**1**answer

198 views

### Is there a formula for the number of rational cuspidal curves in surfaces other than P^2?

Let $M$ be a two dimensional compact complex manifold and $A \in H_2(M, \mathbb{Z})$
a fixed homology class. Define a rational curve in $M$ to be $\textit{1-cuspidal}$ if the singularities of the ...

**4**

votes

**1**answer

297 views

### Quadrics and Moduli Spaces

It is well known that $\overline{M}_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret ...

**6**

votes

**1**answer

239 views

### Is there an analog of compactified moduli spaces(/stacks) for smooth manifolds?

This question has been inspired by an answer to the question Reference request: Topology on the space of smooth compact submanifolds; I've asked it in a comment to that answer but then decided to make ...

**0**

votes

**0**answers

173 views

### projective map from $\overline{\mathcal{M}}_{0,n}$

Suppose I have a morphism $f:\overline{\mathcal{M}}_{0,n} \to \mathbb{P}^N$ birational onto its image, and I know exactly what $F$-curves are contracted (or "dually", what divisors are contracted). ...

**4**

votes

**2**answers

389 views

### How to flip one triangulation on a surface into another

Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$.
Is there an algorithm which takes two ...

**2**

votes

**1**answer

141 views

### What is the value of this hyperelliptic Hodge-type integral?

Consider the moduli space
$$
\overline{M}_{0,4}(B\mathbb{Z}/2)
$$
This has virtual (and real) dimension one. In a certain sense this moduli space paramaterizes "genus 1 hyperelliptic curves"; that is, ...

**1**

vote

**1**answer

190 views

### Isotriviality: two definitions

Consider a proper flat morphism of $k$-schemes ($k$ is an algebraically closed field) $ f:X\longrightarrow\mathbb P^1_k$ such that every fiber $X_p$ for $p\in\mathbb P^1_{\mathbb C}$ is a reduced ...

**2**

votes

**2**answers

244 views

### sanity check about a morphism from a stack to its coarse moduli space

Let $Y(3)$ be the fine moduli space (say, over $\mathbb{C}$) representing elliptic curves equipped with a full level 3 structure. Abstractly, there are 24 such structures for any elliptic curve, but ...

**6**

votes

**1**answer

266 views

### Is there a Riemann existence theorem for orbifolds?

For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified ...

**4**

votes

**1**answer

435 views

### Results about moduli of surfaces

There are early successes of the moduli theory
- the construction and compactification of the moduli spaces of curves $\overline{\mathcal{M}}_g$ .
I want to study about the moduli of algebraic ...

**6**

votes

**2**answers

322 views

### Moduli spaces of connections as representation spaces

It is well known that the moduli space of flat connections over a closed manifold $M$ can be identified with the representation space $Hom(\pi_1(M), G) / G$. Furthermore, Atiyah and Bott (1983) showed ...

**1**

vote

**1**answer

215 views

### Number of singular fibers in families of hypersurfaces

Consider the projection map
$$\pi: X = V(t_0 f + t_1 gh) \to \mathbf P^1,$$
where $[t_0: t_1]$ are the homogeneous coordinates on $\mathbf P^1$, $f=f(x_0, \dots, x_n)$ is a homogeneous polynomial of ...