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.

Filter by
Sorted by
Tagged with
59 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 ...
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
43 votes
2 answers
5k 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 ...
Chris Gerig's user avatar
  • 17.1k
41 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
40 votes
13 answers
15k views

What is a good introductory text for moduli theory?

Hi,everyone. I am looking for an introductory textbook on moduli theory,about the background on algebraic geometry,I have read Hartshorne chapter1~4. could you please show some good books or roadmap ...
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
  • 1,607
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
  • 6,810
35 votes
5 answers
3k views

Intuition behind moduli space of curves

For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the ...
Dinakar Muthiah's user avatar
34 votes
4 answers
3k views

Curves which are not covers of P^1 with four branch points

The following interesting question came up in a discussion I was having with Alex Wright. Suppose given a branched cover C -> P^1 with four branch points. It's not hard to see that the field of ...
JSE's user avatar
  • 19.1k
33 votes
3 answers
3k views

Help motivating log-structures

I'm currently reading a thesis that uses log-structures. I should mention that this is my first encounter with them, and the thesis (as well as my expertise) is scheme-theoretic (in fact stack-...
32 votes
2 answers
3k 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 ...
Jeffrey Giansiracusa's user avatar
31 votes
6 answers
3k views

What can you do with a compact moduli space?

So sometime ago in my math education I discovered that many mathematicians were interested in moduli problems. Not long after I got the sense that when mathematicians ran across a non compact moduli ...
solbap's user avatar
  • 3,938
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
29 votes
3 answers
3k views

Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches?

The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is an isomorphism in ...
Kevin H. Lin's user avatar
  • 20.7k
29 votes
0 answers
739 views

Why do H_4 and M_4 have the same virtual Euler characteristic?

Here's a funny coincidence: The virtual (or "orbifold") Euler characteristic of $\mathcal M_g$ is known by the work of Harer and Zagier: one has $\chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)$. Now ...
Dan Petersen's user avatar
  • 39.2k
28 votes
1 answer
2k views

How were moduli spaces defined before functors?

People today in algebraic geometry will typically define a moduli space to be the space which represents the functor of families of whatever object they are interested in studying. However, I am ...
user332's user avatar
  • 3,878
27 votes
6 answers
4k 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 ...
Francesco's user avatar
  • 281
27 votes
2 answers
1k views

Conformal blocks vector bundles on $\overline{M}_{g}$ in terms of generalized theta functions?

Conformal field theory uses representation theory to produce various vector bundles on the Deligne-Mumford compactified moduli spaces $\overline{M}_{g}$ and $\overline{M}_{g,n}$, known as bundles of ...
Noah Giansiracusa's user avatar
27 votes
1 answer
1k 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 $M_g$, ...
Dan Petersen's user avatar
  • 39.2k
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
  • 39.2k
26 votes
1 answer
1k views

Why there is a Quot-scheme, not a Sub-scheme?

Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably ...
evgeny's user avatar
  • 1,980
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
2 answers
1k views

Different interpretations of moduli stacks

I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to ...
Randy Brown's user avatar
  • 1,386
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
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
24 votes
1 answer
644 views

Rim hook decomposition and volume of moduli spaces

I did some computer experiments, counting the number of rim-hook decompositions (aka border-strip decompositions) of rectangles of shape $2n \times n$, where each strip has size $n$. Here are 12 of ...
Per Alexandersson's user avatar
23 votes
2 answers
2k views

How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that ...
Yuhang Chen's user avatar
  • 1,099
23 votes
3 answers
1k views

What's "bad" about unstable sheaves?

To construct a (coarse or fine) moduli space that is separated, one usually throw away some class of the object in question. For moduli of sheaves people talk about (semi-)stability. A coherent sheaf $...
temp's user avatar
  • 1,990
23 votes
1 answer
2k views

Do hyperKahler manifolds live in quaternionic-Kahler families?

A geometry question that I thought about more seriously a few years ago... thought it'd be a good first question for MO. I'm aware that there are a number of Torelli type theorems now proven for ...
Marty's user avatar
  • 13.1k
22 votes
3 answers
1k views

A historical question: Hurwitz, Luroth, Clebsch, and the connectedness of $\mathcal{M}_g$

The connectedness of the moduli space $\mathcal{M}_g$ of complex algebraic curves of genus $g$ can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the ...
JSE's user avatar
  • 19.1k
21 votes
5 answers
3k views

Is a 'generic' variety nonsingular? Or singular?

I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could ...
Saul Glasman's user avatar
  • 2,148
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
1 answer
2k views

Why not add cuspidal curves in the moduli space of stable curves?

Let $\mathcal{M}_{g,n}$ be the moduli space (stack) of stable smooth curves of genus $g$ with $n$ marked points over $\mathbb{C}. $ It's known that by adding stable nodal curves to $\mathcal{M}_{g,n}$,...
Yuhang Chen's user avatar
  • 1,099
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 ...
20 votes
2 answers
2k 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 ...
André Henriques's user avatar
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
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
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
19 votes
0 answers
501 views

Other examples of computations using transfer of structure from the chains to the homology?

There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...
Jim Stasheff's user avatar
  • 3,840
18 votes
2 answers
2k views

Are Jacobians principally polarized over non-algebraically closed fields?

How does one define the Torelli map $M_g \to A_g$ of moduli stacks? On geometric points a curve maps to its principally polarized Jacobian. So what I am asking is: if I have a curve $C$ over a non-...
David Zureick-Brown's user avatar
18 votes
1 answer
2k views

Does the moduli space of genus three curves contain a complete genus two curve

Inspired by the question Does the moduli space of smooth curves of genus g contain an elliptic curve and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth ...
18 votes
3 answers
2k views

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
  • 9,937
18 votes
1 answer
786 views

Do mapping classes have gonality?

(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.) The hyperelliptic mapping class group is (by ...
JSE's user avatar
  • 19.1k
17 votes
2 answers
2k views

Why is one interested in the mod p reduction of modular curves and Shimura varieties?

Why is one interested in the mod p reduction of modular curves and Shimura varieties? From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the ...
user avatar
17 votes
6 answers
4k views

Why and how are moduli spaces of (semi)stable vector bundles well-behaved?

The slope of a vector bundle $E$ is defined as $\mu(E) = \deg(E)/\mathrm{rank}(E)$. Then a vector bundle $E$ is called semistable if $\mu(E') \leqslant \mu(E)$ for all proper sub-bundles $E'$. It is ...
Kevin H. Lin's user avatar
  • 20.7k
17 votes
4 answers
2k views

What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve?

Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well known, cool formula ...
James O's user avatar
  • 445
17 votes
4 answers
4k views

Soft question: beginners reference to moduli spaces

What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces? (Possibly introducing stacks also)? I'm looking for something which really gets the pictures ...
ABIM's user avatar
  • 5,019
17 votes
3 answers
3k views

Is there an algebraic construction of the Quillen (determinant) Line Bundle?

Let's consider the moduli space of representations of $\pi=\pi_1(\Sigma)$ (a surface group) into $G$ (a lie group). Call this $X=\operatorname{Hom}(\pi,G)$, and let $Y=\operatorname{Hom}(\pi,G)/\\!/G$...
John Pardon's user avatar
  • 18.3k
17 votes
5 answers
2k views

Arithmetic dynamics and dynamics on moduli spaces

The following question is more of a request for pointers to suitable literature on introductory material for arithmetic dynamics and dynamics on moduli spaces. In my dissertation, I have been ...
user avatar
17 votes
1 answer
717 views

Special fiber of $X(p)$ in characteristic $p$

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_0(p)$ be the fine moduli space representing ...
Emmanuel Lecouturier's user avatar

1
2 3 4 5
18