Questions tagged [moduli-spaces]

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 moduli space.

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Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
Tom Copeland's user avatar
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30 votes
3 answers
4k views

When is a classification problem "wild"?

I hope someone can point me to a quick definition of the following terminology. I keep coming across wild and tame in the context of classification problems, often adorned with quotes, leading me to ...
José Figueroa-O'Farrill's user avatar
16 votes
2 answers
2k views

Good introductory references on moduli (stacks), for arithmetic objects

I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and ...
k.j.'s user avatar
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58 votes
11 answers
20k views

What are some open problems in algebraic geometry?

What are the open big problems in algebraic geometry and vector bundles? More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...
16 votes
2 answers
3k views

Is the Torelli map an immersion?

The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see this question for a description which works for families....
David Zureick-Brown's user avatar
14 votes
3 answers
3k views

Existence of fine moduli space for curves and elliptic curves

For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...
Anweshi's user avatar
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38 votes
1 answer
2k views

Why is there no Brauer scheme?

Let $X$ be a proper scheme over a base field $k$ (one could consider more general settings, but I am primarly interested in a "geometric" situation with $k$ being algebraically closed). Then the ...
user25309's user avatar
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24 votes
2 answers
4k views

Why is the degree:rank ratio of a vector bundle called its "slope"?

Whenever one studies moduli spaces of vector bundles on curves, one of the first things to be introduced is the "slope" of a vector bundle, i.e., its degree:rank ratio. Is there a nice (preferably ...
Charles Staats's user avatar
19 votes
4 answers
2k views

Details for the action of the braid group B_3 on modular forms

I'm reading Terry Gannon's Moonshine Beyond the Monster, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an ...
Qiaochu Yuan's user avatar
12 votes
0 answers
983 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 \text{...
jlk's user avatar
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12 votes
1 answer
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Which Riemann surfaces arise from the Riemann existence theorem?

The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi_1(X \backslash \Delta) \to S_d$ ...
skupers's user avatar
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11 votes
1 answer
2k views

Seiberg-Witten theory on 4-manifolds with boundary

What generalizations of Seiberg-Witten theory to 4-manifolds with boundary do exist? I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to ...
J Fabian Meier's user avatar
57 votes
7 answers
8k 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 ...
user avatar
40 votes
1 answer
18k views

What is "Teichmüller Theory" and its history?

What is "Teichmüller Theory"? What part has been worked out / foreseen by O. Teichmüller himself and what is further development? Is there some current work which might be considered as continuation/...
Alexander Chervov's user avatar
39 votes
2 answers
3k 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} \frac{n^...
Max's user avatar
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26 votes
3 answers
2k views

Which mapping class group representations come from algebraic geometry?

Let $\Gamma_g$ be the mapping class group of a closed oriented surface $\Sigma$ of genus $g$. There is a natural surjection $t \colon \Gamma_g \to \mathrm{Sp}(2g,\mathbf Z)$ which sends a mapping ...
Dan Petersen's user avatar
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25 votes
9 answers
13k views

Teichmuller Theory introduction

What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?
24 votes
3 answers
5k views

Conceptual understanding of the Gross-Zagier theorem.

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
Regenbogen's user avatar
  • 1,407
21 votes
4 answers
3k views

Betti numbers of moduli spaces of smooth Riemann surfaces

Where can I find a list of the known Betti numbers of the moduli spaces $\mathcal{M}_{g,n}$ of genus $g$ Riemann surfaces with $n$ marked points? I need it to cross check results by an implemented ...
domenico fiorenza's user avatar
21 votes
2 answers
2k views

Applications of derived categories to "Traditional Algebraic Geometry"

I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
19 votes
3 answers
4k views

Moduli space of K3 surfaces

It is known that there exists a fine moduli space for marked (nonalgebraic) K3 surfaces over $\mathbb{C}$. See for example the book by Barth, Hulek, Peters and Van de Ven, section VIII.12. Of course ...
Andrea Ferretti's user avatar
19 votes
3 answers
2k views

What is the DGLA controlling the deformation theory of a complex submanifold?

Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. ...
Braxton Collier's user avatar
15 votes
1 answer
1k views

Number of curves over a finite field

Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$? In other words does there exists a formula for the number of rational points ...
Puzzled's user avatar
  • 8,832
15 votes
4 answers
3k 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?
Mohammad Farajzadeh-Tehrani's user avatar
15 votes
3 answers
1k views

Are there graph models for other moduli spaces?

Recall that a ribbon graph is a graph with a cyclic ordering at each vertex and such that each vertex has valence greater than or equal to 3. This cyclic ordering exactly gives one the information to ...
skupers's user avatar
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15 votes
1 answer
2k views

Moduli space of motives vs moduli space of varieties

A (projective) abelian variety $A$ over the complex numbers is determined by $H^1(A,\mathbb{Z})$ together with its Hodge structure and polarization. This miracle means that one can parametrise ...
eric's user avatar
  • 829
13 votes
3 answers
1k views

The use of Hall algebras in physics

I once read a statement (not memorized precisely) that a certain physics quantity between two states of charge $d_1$ and $d_2$ respectively could be computed by running over the states of charge $d_1+...
Xiao Xinli's user avatar
12 votes
2 answers
2k 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. ...
Qfwfq's user avatar
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11 votes
1 answer
878 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 ...
Jack's user avatar
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11 votes
1 answer
712 views

What is the Brauer group of the moduli space of (p.p.) abelian varieties?

What is the Brauer group of the moduli space of principally polarized abelian varieties of a given dimension? I am primarily interested in the "open" moduli space, i.e. not a compactification. The ...
Felipe Voloch's user avatar
9 votes
3 answers
1k views

Stable graphs: Feynman diagrams and Deligne-Mumford space

I do not know very much about quantum field theory, but I have seen, in my reading, that stable graphs can appear in QFT in the form of, I think, Feynman diagrams. By stable graph I mean a "graph with ...
Kevin H. Lin's user avatar
  • 20.7k
9 votes
1 answer
567 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 ...
El Nino's user avatar
  • 93
9 votes
1 answer
1k views

Non-representable functor, representable on locally Noetherian schemes?

What is an example of a functor $F : \mathbb{C}\text{-Sch.} \to \text{Sets}$ with the property that the restriction of $F$ to locally Noetherian $\mathbb{C}$-schemes can be represented by a locally ...
jlk's user avatar
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8 votes
1 answer
1k views

Progress on Bondal–Orlov derived equivalence conjecture

In their 1995 paper, Bondal and Orlov posed the following conjecture: If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent ...
mathphys's user avatar
  • 305
8 votes
1 answer
636 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 + ...
Will Sawin's user avatar
  • 135k
7 votes
1 answer
2k views

Dualizing sheaf on singular curves

I am trying to understand the stabilization map, which takes a prestable curve (a curve with some marked points, and at worst nodal singularities) and returns a stable curve (a curve with some marked ...
Kevin H. Lin's user avatar
  • 20.7k
7 votes
0 answers
492 views

Compactification of the moduli space of Kähler manifolds with negative constant scalar curvatures

Moishezon compactification is very important in the study of the moduli space of varieties which admit canonical metrics. Moishezon showed that any non-projective Moishezon manifold $X$, after a ...
user avatar
7 votes
1 answer
888 views

Modular curve X(2)

Let $\mathfrak{M}(2)$ be the algberaic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the two level structure and let $X(2)$ be the coarse moduli space of $\mathfrak{M}(2)$ ($X(...
Adel BETINA's user avatar
  • 1,046
7 votes
1 answer
365 views

Selberg Zeta Function and Fenchel-Nielsen Coordinates

According to Uniformization theorem every compact Riemann surface $\Sigma$ of genus $g\ge2$ is isomorphic to a space that can be obtained by the action of a Fuchsian group on upper half plane $\mathbb{...
QGravity's user avatar
  • 969
7 votes
0 answers
216 views

For all schemes w/Hilbert polynomial P, exists $m_P$ s.t. no higher cohomology, $I(k)$ generated by globally sections, multiplication is surjective

Consider the following theorem. For every polynomial $P$, there exists an integer $m_P$ such that for all ideal subsheaves $I \subset \mathcal{O}_{\mathbb{P}^n}$ with Hilbert polynomial $P$ and every ...
Tom Berrett's user avatar
6 votes
2 answers
727 views

Questions on theorem in Deligne-Mumford's '69 Paper: $\omega_C^n$ is very ample $n\geq 3$

I'm working through the details of Deligne and Mumford's 69' paper, "The Irreducibility of the Space of Curves of Given Genus", and I had a few quick questions: 1) On p. 77, they claim that for $x$ a ...
HNuer's user avatar
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6 votes
2 answers
1k views

Uniformizations of the bordered/punctured Riemann surfaces

The uniformization theorems of Riemann surfaces state that any Riemann surface can be constructed by an action of some group on some space. It is quite hard to find materials relating different ...
QGravity's user avatar
  • 969
5 votes
1 answer
190 views

Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface

If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups: $$\text{Mod}(\mathcal{R}')\longrightarrow \...
QGravity's user avatar
  • 969
5 votes
0 answers
417 views

Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
Dan Petersen's user avatar
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5 votes
0 answers
533 views

connectedness of moduli space of Calabi-Yau 3-folds by symplectic surgery theory

"Motto" Moduli space of Calabi-Yau varieties can be connected by using Symplectic surgery theory. Miles Reid’s Fantasy:“There is only one Calabi-Yau space” i.e "All CY connected through conifold ...
user avatar
4 votes
0 answers
229 views

Virtual fundamental class of Moduli space of stable maps in genus 1

What is the virtual fundamental class of $\overline{M}_{1,n}(\mathbb{P}^2,d)$? In general the virtual fundamental class is difficult to compute I guess. But if you look at Proposition 2.5 of https://...
Chitrabhanu's user avatar
3 votes
0 answers
140 views

Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?

Following up on this question which received a negative answer, I wonder if something weaker is true. We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...
Jef's user avatar
  • 949
3 votes
1 answer
357 views

Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ ...
user51764's user avatar
  • 139
3 votes
0 answers
146 views

Examples of subspaces singled out by modular forms

I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature. To be more clear let me explain the example I have in mind. Let $N\...
Bear's user avatar
  • 231
3 votes
0 answers
351 views

Universal moduli space of rank two vector bundles over a curve

Let $\mathcal M_C$ denote the moduli space of rank two vector bundles over a smooth proper curve $C$ over an algebraically closed field $k$ of characteristic zero. Let $k(t)$ denote the function ...
John Pardon's user avatar
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