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 ...

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2
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81 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
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0answers
85 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
0answers
55 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
0answers
114 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
0answers
236 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
0answers
83 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
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0answers
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
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0answers
95 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
0answers
110 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 ...
4
votes
1answer
171 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
2answers
440 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
1answer
186 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 ...
1
vote
0answers
95 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
1answer
85 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
1answer
298 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
0answers
91 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
1answer
178 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
0answers
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
0answers
63 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
4answers
319 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
1answer
253 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
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0answers
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
0answers
262 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
1answer
109 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
1answer
274 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
0answers
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
0answers
96 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
0answers
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 $\mathscr{M}$ ...
2
votes
1answer
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
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163 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
0answers
282 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
1answer
339 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
0answers
217 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
0answers
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
2answers
875 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
0answers
177 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 ...
2
votes
0answers
172 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
2answers
193 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
1answer
240 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
1answer
143 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
1answer
801 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
0answers
199 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
2answers
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
1answer
325 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 ...
3
votes
0answers
116 views

When does a “universal” quot scheme exist?

Suppose $M$ is a moduli space of semistable sheaves on a projective variety $X$. Let $v$ be some the discrete invariants. I would like to form a space $\mathcal Q(v) \rightarrow M$, where the fiber ...
11
votes
1answer
236 views

$\pi_1$ of the moduli of G-bundles on elliptic curves and the double affine braid group

For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, the fundamental group of $\mathfrak{h}_\text{reg}/W$ (where $\mathfrak{h}$ is the Cartan subalgebra, $\mathfrak{h}_\text{reg}$ is the subset ...
3
votes
0answers
112 views

Dimension of the singular locus of $\mathcal M_X(r,d)$

Let $X$ be an algebraic smooth curve of genus $g$ over $\mathbb C$, and let $\mathcal M_X(r,d)$ (resp $\mathcal M_X^0(r)$) be the moduli space of vector bundle of rank $r$ and degree $d$ (resp. with ...
4
votes
1answer
192 views

Existense of semi-stable vector bundles on smooth curves in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic and $X$ be a smooth projective curve over $k$ of genus $g \ge 2$. Fix a polarization $L$ on $X$. Does there exist a semi-stable ...
3
votes
0answers
95 views

Choosing a group action to do GIT of hypersurfaces

When studying GIT stability of hypersurfaces $d$ of $\mathbb P^n$ we look at the Hilbert Scheme $H=\mathbb P^N$ parametrizing homogeneous polynomials $f_d(x_0,\ldots,x_n)$ of degree $d$. There is ...
15
votes
1answer
292 views

Moduli space of boundary maps with prescribed chain and homology groups?

Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every ...