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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How to characterize good "models" of a category
Let ${\bf Cat}$ denote the category of small categories. Recall that for a category $\mathcal{C}$ and a functor $F\colon\mathcal{C}\to{\bf Cat}$, the Grothendieck construction of $F$, which I'll ...
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Question related to the moduli space of Riemann surfaces and a fibration
If $M_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map:
$M^1_{g} \...
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Homology dimension of the mapping class group of a surface with boundary
There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is
$H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except $(...
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Curious propositon in "Les schemas de modules de courbes elliptiques"
Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation):
(II ...
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How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?
How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...
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Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is hausdorff
Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is Hausdorff? Note: here $\bar{M_{g,n}}$ is not the Deligne-Mumford space in the usual algebraic ...
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Does the Deligne-Mumford space module $S_{n}$ action have a fundamental chain?
Does the Deligne-Mumford space (without ordering for marked points) $\bar M_{g,n}/S_{n}$ has fundamental chain in signular simplicial chains? (because I read Costello's paper GW potential to TCFT, as ...
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Central Yang-Mills connections, and flat connections with prescribed holonomy
Let $X$ be $\Sigma^g$ which is the Riemann surface of genus $g$, and consider a trivial $G$-bundle over it.
1) In this $2$-d setting, the space of Yang-Mills central connections is the set of ...
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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). ...
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projective subvarieties of the moduli space of abelian varieties
I know that the fibre of $A_{g,n}$ over $\mathbf{F}_p$ is quasi-projective (of what dimension?). Can one exhibit some smooth projective subvarieties of high dimension in it? What are references for ...
user19475
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The fibers of M_{g,n} \to M_g and the Fulton-MacPherson compactification
Let $g \geq 2$, and consider the moduli space $\bar M_{g,n}$ of stable n-pointed curves of genus g. There is a natural forgetful map to $\bar M_g$, which forgets the markings and contracts any ...
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Moduli of extensions of modules
Given two modules $M$ and $N$ there is a nice scheme parametrizing extensions
$0 \rightarrow M \rightarrow E \rightarrow N \rightarrow 0$
namely $\operatorname{Ext}^1(N,M)$ or, leaving out the trivial ...
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Picard Groups of Moduli Problems
First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible.
I'm told that for $g\geq 2$ it is ...
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Are the arithmetic genera of Cohen-Macaulay curves in a fixed homology class bounded?
Let X be a smooth projective variety over the complex numbers. Recall that a Cohen-Macaulay curve is a one-dimensional closed subscheme without embedded or isolated points (fat components are allowed)....
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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 ...
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A-infinity structure on the ribbon graph complex and more general graph complexes
Moduli spaces of curves (with nonempty boundary or at least one marked point) admit cell decompositions in which the cells are labelled by ribbon graphs. In fact, the moduli space of normalised ...
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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 ...
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Reference request: Moduli spaces of bundles over singular curves
I would like to know some reference (articles, books...) about any kind of moduli spaces of any of the following objects:
vector bundles
torsion-free sheaves
principal bundles
parabolic bundles
over ...
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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 ...
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Gauge theory construction of moduli of vector bundles
Given a smooth projective complex variety $X$, instead of using Mumford's GIT to construct the moduli of rank $n$ topologically trivial vector bundles, we can also take the gauge theory approach.
To ...
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Dualizing sheaf of reducible variety?
Sorry for my poor English.
Let $X$ be a reducible projective variety.
My question is:
How can I compute the dualizing sheaf of $X$ and express it in an explicit way?
Is there a method to get ...
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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 ...
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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 ...
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Versality in deformation theory vs. versality in moduli spaces
As I mentioned before, I'm a novice at deformation theory. I was wondering if the definition of versality in deformation theory is related to the versality in moduli spaces:
Deformation theory
"...
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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 ...
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Family of Enriques surfaces and Grothendieck-Riemann-Roch
Currently I'm studying the article Moduli of Enriques surfaces and Grothendieck-Riemann-Roch by Pappas.
I am particularly interested in how he applies the GRR.
Q1. What is meant by a "family of ...
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looking close at an example of Moduli space of curves
I will state a very specific case: genus 5. Though it's particular, it admits a generalization to $M_g$, and I think reflects the nature of a general stratification of $M_g$.
It is known that if you ...
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MNOP conjecture
Let $X$ be a smooth, projective, Calabi-Yau 3-fold (CY makes the exposition more elegant, I don't think it is necessary).
To define Gromov-Witten invariants, we consider moduli spaces of stable ...
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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 ...
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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 ...
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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 ...
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Some questions on the intersection theory on a Hilbert scheme of points of a surface.
If $\Sigma$ is a smooth complex curve in a smooth projective surface $X$, then we can consider the homology class represented by $\Sigma^{[n]} \subset X^{[n]}$. $\ \ $ Where, $X^{[n]}, \Sigma^{[n]}$ ...
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Trigonal loci in Teichmueller spaces
Since my previous question
Hyperelliptic loci in Teichmueller spaces
resulted in two quick and helpful replies, let me ask another question in a similar vein:
A smooth compact complex curve is ...
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Hyperelliptic loci in Teichmueller spaces
Let ${\cal M}_g$ be the moduli space of smooth complex genus $g$ curves, let ${\cal H}_g\subset {\cal M}_g$ be the hyperelliptic locus and set ${{\cal H}}'_g$ to be the preimage of ${\cal H}_g$ in the ...
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Proper definition of a moduli problem
This question arose after I thought about Ben Webster's comments to this question.
There he asked me what was my definition of a moduli problem. When I came to think of it, I never saw a precise ...
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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 ...
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(nontrivial) isotrivial family of elliptic curves
I think it should be a standard procedure to construct such things, can anyone give a reference or give a hint? Can this be done over any base scheme?
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moduli space and modularity
I recently realized some kind of analogy when considering modularity results (such as the modularity of elliptic curves over Q). The analogy comes from algebraic groups. Take one point (say, the ...
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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 ...
user19475
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Introductory text for the non-arithmetic moduli of elliptic curves
I'm looking for an introduction to the non-arithmetic aspects of the moduli of elliptic curves. I'd particularly like one that discusses the $H^1$ local system on the moduli space (whether it's $Y(1)$ ...
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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 ...
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functorial meaning of irreducibility of a moduli space
Iasked me the question what the interpretation of the irreducibility of a moduli space is for the functor it represents. For proper, there is the valuative criterion and for (formally) smooth, there ...
user19475
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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 ...
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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 ...
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When is a coarse moduli space also a fine moduli space?
Given a moduli problem, it appears that nonexistence of automorphisms is a necessary condition for existence of a fine moduli space(is this strictly true?).
In any case, assuming the above, what ...
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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?
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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 ...
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Moduli spaces of coherent sheaves on K3s
Reading 2007 paper A tour of theta dualities on moduli spaces of sheaves by Alina Marian and Dragos Oprea.
Why is any moduli space of coherent sheaves on a K3 surface deformation equivalent to a ...
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Coarse moduli spaces over Z and F_p
I would like to know to what extent it is possible to compare fibers over $\mathbb{F}_p$ of coarse moduli spaces over $\mathbb{Z}$, and coarse moduli spaces over $\mathbb{F}_p$. I ask a more precise ...
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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....
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