Questions tagged [moduli-space]

A Moduli space is a space in algebraic geometry whose points are geometric objects or isomorphism classes of these kinds of objects.

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Why use orientation-preserving diffeomorphism (instead of all diffeo's) in the construction of the moduli space of a Riemannian manifold

The question is basically in the title, but I want to make it more precise: Given an oriented Riemannian 2-manifold $\Sigma$ one can take a quotient of the set $$ \mathcal{M}_+(\Sigma)=\{~c~\mid (\...
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What is an instanton? (On a complex surface or a differentiable 4-manifold )

The question is as in the title. I have browsed online (Wikipedia, etc) and while they do give me the definition, it gets a bit too much physics-y for me. Therefore I would appreciate it if someone ...
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The stack classifying non-oriented triangles is equivalent to a quotient stack $[\widetilde{T}/S_3]$

I come across problems when I try to compute the stack classifying non-oriented triangles. My reference is the book Algebriac Stacks ( K. Behrend, B. Conrad, D. Edidin, B. Fantechi, W. Fulton, L. ...
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Singularities of arithmetic surfaces

I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves. The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
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Uniqueness of model of curve over DVR

I understood that properness of $\overline{\mathcal{M}_{g,n}}$ tells us that for a fixed field $K$, a curve $C$ over $K$ and a DVR $V$ with fraction field $K$, there is a unique curve over $V$ with ...
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Exercise 1.1c in Hartshorne's Deformation Theory: Is this family of conics flat?

In my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ($k$ is ...
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determinant line bundle of ASD moduli space

For a principal bundle $P$ and the space of irreducible connection $\mathcal{A}^*$. We consider the configuration space $\mathcal{B}^*=\mathcal{A}^*/\mathcal{G}$. The book Instantons and Four-...
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calculate the dimension of moduli space

I'm learning gauge theoretic topics about 4-manifolds, and I get stuck when I try to calculate the dimension of ASD moduli space. For an oriented closed 4-manifold $M$, we first fix a riemannian ...
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local model of gauge theoretic moduli space

I'm a beginner of gauge theory and I find that most materials state the theorem(without proof) that: Given a principal bundle $P$, for a connection $A$ over $P$, a isotopy group $G_A$ of $A$ consist ...
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Compactification of $M_{1, 1}$

Let $M_{1, 1, \mathbb{Z}}$ be a moduli stack of elliptic curves. Denote $M_{1, 1, k}$ its base change to a field $k$. We have the map $j: M_{1, 1, k} \to \mathbb{A}^1_{k}$ which realizes the affine ...
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Morphism between Hom schemes.

When $X$ and $Y$ are projective schemes then there is a quasi-projective scheme called Hom-scheme which represents the functor $T\mapsto \text{Hom}(X\times T, Y)$. Let's denote the Hom-scheme by $\...
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Moduli space of connection on line bundle

I want to show that for a $U(1)$ bundle $P$ over a connected smooth 4-manifold $X$, the moduli space of Yang-Mills connection over $P$ is the torus $H^1(X,\mathbb{R})/H^1(X,\mathbb{Z})$. Now I reduce ...
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Assumptions on a scheme X needed to construct the Picard scheme $\operatorname{Pic}(X)$

Let $X$ be a scheme. I believe that Grothendieck was the first to put a scheme structure on the Picard group $\operatorname{Pic}(X)$, where he assumed that X is projective and reduced. Then Mumford ...
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Slope of tensor product of coherent sheaves

Let $X$ be a projective normal variety with a fixed very ample bundle $\mathscr{O}(1)$, and $E$, $F$ torsion-free coherent sheaves on $X$. In the page 29 of ``The Geometry of Moduli Spaces of Sheaves'...
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What do we know about the space of homomorphisms of semistable vector bundles?

Let $X$ be an algebraic curve over $\mathbb C$ (feel free to add smoothness or other usefule properties). Fix degrees $d_1,d_2$ and ranks $r_1,r_2$. The slopes are $\mu_i=d_i/r_i$. Assume $\mu_1>\...
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Étale local universal sheaves

Moduli functors for (flat families of) semistable coherent sheaves over a projective scheme are not representable functors, which is to say that no global universal family exists. However one can ...
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Moduli space of a sphere

What does moduli space mean in simple terms? What is the moduli space of a Riemann sphere? What is the moduli space of an n-marked sphere? Is there a very basic reference for moduli spaces that would ...
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Reference request: Katz modular forms modulo $p$ at cusps vs classical modular forms modulo $p$ at cusps

I, like the author of this post, am severely lacking the background to make the connection between reducing modular forms' $q$-expansions modulo $p$ at various cusps, and $q$-expansions of modular ...
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Is a coarse moduli space whose objects have no automorphisms also fine?

We work over the complex numbers $ k = \mathbb{C} $. Let $ F $ be a moduli problem, i.e. a contravariant functor $ F : \text{Sch}/k \rightarrow \text{Sets} $. Suppose that $ M $ is a coarse moduli ...
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Involution of Legendre Family

I'm trying to understand why automorphisms of elliptic curves lead to non-existance of the moduli space (as a variety). Some notes suggest to consider the following example: Consider the Legendre ...
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What is the moduli space $M _{0,5}$?

Let $M_{0,n}$ denote the moduli space which consists of genus 0 non singular projective curves with n distinct marked points upto marked point isomorphism. So $M_{0,3}$ is a singleton set as any 3 ...
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How to construct the schemes $\operatorname{Hom}_S(X, Y)$ and $\operatorname{Isom}_S(X, Y)$ from the Hilbert scheme?

Let $S$ be a noetherian scheme. Grothendieck [1] constructed for any projective $S$-scheme $X$ the Hilbert scheme $\operatorname{Hilb}_{X/S}$, which represents the functor $$T \mapsto \underline{\...
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Moduli space of special Lagrangians

I'm currently reading Auroux's Mirror Symmetry and T-duality in the Complement of the Anticanonical Divisor and Special Lagrangian Fibrations, Wall-crossing, and Mirror Symmetry back and forth. I'm ...
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Teichmuller space of the 4-punctured sphere

I'm a bit confused working with Teichmuller space at the moment. Let's think of Teichmuller space as the space of holomorphic/conformal structures on a surface, up to diffeomorphisms isotopic to the ...
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What does it mean for a space to "carry a natural integration"?

In a ncatlab article, I came across two sentences (on the image below) that are not so clear to me. What does it mean that the compactifications "carry natural integration"? I guess it has ...
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Notation in Katz-Mazur Arithmetic Moduli of Elliptic Curves

A moduli problem $\mathcal{P}$ is a contravariant functor $\mathbf{Ell}\to\mathbf{Set}$. The objects of $\mathbf{Ell}$ are arrows $E\to S$ from an elliptic curve $E$ to a varying base scheme $S$. The ...
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Is the moduli stack of stable maps functorial?

Let $f:X\rightarrow Y$ be a morphism where $X$ and $Y$ are smooth projective varieties over $\mathbb{C}$. Does $f$ induce a well-defined "pushforward" morphism $\overline{\mathcal{M}}_{g,n}(...
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Configuration spaces to moduli spaces

In Segal's paper on Mapping Configuration spaces to moduli spaces, I'm not understanding what the map $\Phi$ is, explicitly. Also in section 2, he goes on to say $M_{g,2} \simeq BHomeo^{+}(F_{g,2}; \...
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Coarse Moduli Space of Quotient Stack

The following (or maybe a variant) seems pretty well known: if $G$ is a finite group acting on an affine scheme $\text{Spec}A$ which is of finite type over a Noetherian ring $R$, then the map $$[\text{...
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4 votes
1 answer
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moduli space of elliptic curves, $\mathbb{H}/SL(2,\mathbb{Z})$ or $\mathbb{H}/PSL(2,\mathbb{Z})$

I see some (most) people saying the moduli space of elliptic curves is $\mathcal{M}_{1,1}=\mathbb{H}/SL(2,\mathbb{Z})$ which is an ineffective orbifold. But it is also fine to forget the trivial ...
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When is $\operatorname{Spec} A^G \cong (\operatorname{Spec} A)/G$ true?

Let $G$ be a group acting on a ring $A$. I would like to know in which generality we know that $\operatorname{Spec} A^G \cong (\operatorname{Spec} A)/G$. Moreover, when this is true, it also holds for ...
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7 votes
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What does it mean for points of the modular curve $X(N)$ to be "defined over $\mathbb{F}_p$"?

I'm trying to study a collection of elliptic curves over some fixed finite field $\mathbb{F}_p$. By browsing the literature and discussing with my supervisor, it seems like it will be fruitful to ...
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1 vote
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Isomorphism of curves induces isomorphism of moduli spaces of vector bundles on the curves?

Suppose we have an isomorphism of projective curves $f : C \to C'$. Let $M_C(\mu)$ denote the moduli space of slope stable vector bundles on $C$ with slope $\mu$. Do we get an induced isomorphism $F : ...
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Historical significance of moduli spaces

Last semester I took a class on symplectic topology where we proved Gromov's theorem using moduli spaces of spheres. This semester, I am taking algebraic geometry following Hartshorne and after ...
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Quotient of $GL_n$ by unipotent group and maximal torus

Assume we are working over a field $k$ and $G= GL_n$. Then we get the (standard) maximal torus $T$, the unipotent group $U$ and the Borel $B$ of upper triangular matrices. More generally we have we ...
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Stable graph of stable curves

Let $\Gamma = (\mathbf{V},\mathbf{H}, \mathbf{L},g, v,i,\ell)$ be a stable graph. Let $(\mathcal{C},p_1,\cdots,p_n)$ be a stable curve with the dual curve $\Gamma$ and let $(\tilde{\mathcal{C}}_v,(q_h)...
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What is the definition of moduli space, in math vs in physics?

It is easy to find that there are many questions regarding moduli space on MSE: https://math.stackexchange.com/search?q=what+is+moduli+space But it seems to me that this phrase, moduli space, may mean ...
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2 votes
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Applications of the Chinese remainder Theorem to the study of the Hilbert scheme of points and $(\mathfrak{m},l)$-squeezed ideals.

The following construction gives a relation between the Chinese Remainder Theorem (CRT), the Noether nomalization lemma (NNL) and cofinite ideals in finitely generated $k$-algebras. Let $k$ be any ...
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Construction of Hilbert Variety (in Harris' Algebraic Geometry)

My question intends to take up a detail in the construction of the Hilbert variety in Harris' book Algebraic Geometry; A First Course (p 274). Let $S=K[X_0,X_1,..., X_n]$ be the ring of polynomials in ...
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Hyperplane $\Phi_X$ in Chow construction (Algebraic Geometry by Joe Harris)

I'm trying to figure out what Harris wanted to say in following construction, called Chow construction (Algebraic Geometry by Joe Harris, p. 269): The first construction of the parameter space for ...
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-5 votes
2 answers
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Some integers related to the Hilbert scheme of points in the plane. [closed]

This question is related to another question posed on this site. Let me recall the construction: Let $A:=k[x,y]/I$ with $k$ the complex numbers (or any algebraically closed field) and $\dim_k(A)< \...
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Reduction of structure group $GL_n$ of the endomorphism bundle to the centraliser.

I was reading this proof from http://www.numdam.org/item?id=AST_1982__96__1_0 where of structure group of the bundle $EndE$ which is taken to be $GL_n$ is reduced to the centraliser of $GL_n$. The ...
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1 vote
1 answer
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Can we factor this quotient ring.

Let $I \subset \mathbb{C}[x,y]$ be an ideal such that $\dim \mathbb{C}[x,y]/I =n$ for some natural number $n$. For any point $p \in \mathbb{C}^2$ we define $I_p=(x-p_x,y-p_y)^n+I$. I want to prove ...
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4 votes
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Coarse Moduli space of plane cubics

I am studying Joe Harris' Algebraic Geometry: A First Course, the section on Moduli Spaces, pg 278. I am stuck in a subtle point. Harris gives on p 279 an argument why there is no coarse moduli space ...
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Question about the proof that there does not exist a fine moduli space for endomorphisms of $n$-dimensional vector spaces.

I am reading Mumford's 1970 Oslo lecture on moduli theory. In this lecture, he defines a family of n-dimensional vector space endomorphisms over a scheme $T$ to be a pair $(\mathcal{E}, \phi)$ where $\...
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5 votes
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Help with moduli spaces of four-marked spheres

I'm thinking about the moduli space of the four-punctured sphere where some of the removed points are distinguishable and some are indistinguishable. I believe there should be some covering maps ...
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Isomorphism between $\overline{U}_{0,4}$ and the degree $5$ Del Pezzo Surface

Tacitely, I am working over the field of complex numbers! Let $\overline{M}_{0,4}\cong\mathbb{P}^1$ be the compactification of the moduli space of $4$-pointed stable rational curves. The relevant ...
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6 votes
1 answer
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What is Representation of Surface Groups?

I had a question in my mind for a month ago. Mainly I am interested in Hyperbolic Geometry. I found a topic named "Representation Theory of Surface Groups". Let me tell about what is a "...
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Family of stable curves over disc

Let $(C,x)$ be a fixed curve of genus $g \geq 1$ with one marked point and let $D\in C$ be a small disc centered at $x$. Over the punctured disc $D^*$ we have a natural family of smooth curves with ...
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3 votes
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Properness of the coarse moduli map in Keel-Mori theorem.

Given a stack $\mathscr X$ with enough assumptions we obtain a map $\rho: \mathscr X \to X$ to a coarse moduli space. Furthermore, $\rho$ is proper. I do not understand what it means for $\rho$ to be ...
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