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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7
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2answers
238 views

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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2answers
210 views

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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0answers
22 views

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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18 views

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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1answer
98 views

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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61 views

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)...
6
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1answer
224 views

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 ...
2
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0answers
169 views

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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51 views

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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40 views

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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2answers
259 views

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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40 views

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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1answer
170 views

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 ...
3
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1answer
66 views

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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28 views

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 $\...
5
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1answer
50 views

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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1answer
76 views

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 ...
7
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1answer
144 views

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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1answer
34 views

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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36 views

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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Universal maps and moduli spaces

I am now reading this wonderful thesis on moduli spaces by Djounvona. On Proposition 2.5.3 the author says that if you have a moduli functor $\mathcal{M}$ that is represented by an object $M$ in a ...
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Proof verification : Is my modified moduli problem still representable?

I came across a moduli problem which looks very similar to another one appearing in a little lemma by Rapoport and Zink. I tried to prove that mine is also representable, but as a newbie in this topic ...
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1answer
61 views

Sketches of stable 6-pointed curves

I was wondering if I could ask about stable $n$-pointed curves (essentially intersecting copies of $\mathbb{P}^1$ with at least $3$ special points on each copy - these may be either a point of ...
3
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1answer
82 views

Description of Euler Characteristic Using $\operatorname{Ext}^i$'s

I am reading Nakajima's Lectures on Hilbert Schemes of Points on Surfaces, and I am confused about a detail at the top of page 13. He is sketching a proof that $X^{[n]}(=$ Hilbert scheme of $n$ points ...
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0answers
43 views

Embedding of moduli space of J-holomorphic spheres into moduli space of stable maps

Let $(M,\omega)$ be a closed symplectic manifold of dimension $2m$, $J$ be an $\omega$-compatible almost complex structure, and $A\in H_2(M;\mathbb{Z})$ a homology class. Let $n \geq 3$ and denote by $...
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52 views

Two equivalent definitions of the coLie sheaf for a family of elliptic curves [duplicate]

Let $f: E\rightarrow S$ be a family of elliptic curves, i.e. a proper flat morphism of schemes whose geometric fibers are genus $1$ curves, along with a section $e:S\rightarrow E$. Then there is an ...
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23 views

Why polarization in Torelli theorem?

Are there two non-isomorphic compact Riemann surfaces with isomorphic integral Hodge structure on $H^1(-,\mathbb{Z})$? Recall Torelli theorem for Riemann surface: For two compact Riemann surfaces $X,Y$...
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40 views

Dual Graphs of T-fixed Stable Maps

The following screenshot is taken from "Localization of Virtual Classes" paper of Graber and Pandharipande. I don't understand the following highlighted sentences. From those sentences I ...
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45 views

Morphism from Automorphism group of stable curve to automorphism group of its dual graph surjective

I currently study the strata of the moduli stack of algebraic curves and have understood, that for a stable curve with markings $p_1, \dots, p_n$ there is a morphism of groups $Aut(C, p_1, \dots, p_n) ...
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1answer
75 views

Flabby representable sheaves

Let $S$ be a scheme. Consider some representable moduli functor $\mathcal{M}:(Sch/S)^{op}\rightarrow Set$ represented by some scheme $M$. Then for each $V\in (Sch/S)^{op}$, let define $$\mathcal{M}^{...
3
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1answer
76 views

Moduli Space of Tori

I'm looking at an exercise that reads: Problem. Let $A = \begin{bmatrix}a & b\\c & d\end{bmatrix} \in GL(2, \mathbb{C})$ and $\Lambda = \langle z \mapsto z + \omega_1, z \mapsto z + \omega_2\...
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0answers
41 views

Satake compactification for the moduli space of curves of genus $g$

I am using the book Compact Complex Surfaces by Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, Antonius Van de Ven. On p.220, this book says "It follows readily from the projectivity of the ...
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0answers
37 views

What can we say about moduli spaces of sheaves with double the Chern character of known moduli space of sheaves?

Say I know everything about the moduli space $M_G(v)$ of Gieseker (alternatively, slope, Bridgeland, etc.) stable sheaves with Chern character $v$ on a smooth complex projective variety $X$, where $\...
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0answers
33 views

Why is the Orbifold Euler characteristic of $M_{1, 1}$ equal to $\zeta(-1) $

The orbifold euler characteristic of $M_{1, 1}$= $PSL(2, \mathbb{Z}) $/$\mathbb{H}$is equal to -1/12. This is due to the fact that $M_{1,1}$ has one two-cell with $\mathbb{Z_{2}}$ automorphism, two ...
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0answers
127 views

Hilbert scheme of hypersurfaces in Nitsure’s Construction of Hilbert and Quot Schemes

I have a few questions on the construction of Hilbert scheme of hypersurfaces I found recently in Nitin Nitsure’s Construction of Hilbert and Quot Schemes (page 6 part 4) In his paper Nitsure added a ...
5
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2answers
158 views

Relation/Difference between moduli spaces and classifying spaces.

From what I have read so far, a classifying space is a representing object of some (co)representable functor. For example, the $n^\text{th}$ Eilenberg–MacLane space is the classifying space for the $...
9
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2answers
294 views

Coarse moduli space of relative Picard functor for affine line

Consider the relative Picard functor $\mathrm{Pic}_{\mathbb A^1/\mathrm{Spec}(\mathbb C)}$ sending a complex scheme $X$ to $\mathrm{Pic}(X \times \mathbb A^1)/\pi_X^* \mathrm{Pic}(X)$. Since $\mathrm{...
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0answers
64 views

Singularities of moduli spaces $M_g$

This survey paper from Lizhen Ji says: Teichmüller was aware that nontrivial automorphisms of Riemann surfaces caused difficulty in constructing $M_g$ and singularities of $M_g$, and he ...
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41 views

Morphisms between moduli spaces

Assume that I have two moduli spaces $M_1,M_2$ solving two problems $F_1,F_2:C\to Sets$, think for example about $C=Sch/S$. Furthermore I have a morphism $f:M_1\to M_2$ or equivalently a natural ...
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0answers
21 views

Is the product group of finitely many copies of the multiplicative group of positive real numbers a reductive group?

Let G be the multiplicative group of positive real numbers. Is the finite product group $G \times \cdots \times G$ reductive? I am trying to construct the moduli space for some quiver representations ...
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1answer
55 views

the moduli spaces that contain direct sums of line bundles

Let $U_X(n,d)$ be the moduli space of semistable vector bundle of rank $n$ and degree $d$ over a smooth projective curves over the complex numbers. How do I know if $U_X(n,d)$ contains direct sums of ...
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45 views

When is the coarse moduli space of genus $g$ stable nodal curves singular?

Let $\overline{\mathcal{M}}_g$ be the moduli stack of genus $g$ stable nodal curves and let $\overline{M}_g$ denote its coarse moduli space. In 1969, in the paper "The irreducibility of the space of ...
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0answers
28 views

Moduli functor with no coarse moduli space

I have the following problem. Let $\mathcal{M}$ be the set of isomorphism classes of invertible complex 3$\times$3 matrices. If we have a variety $X$, a family over $X$ is a matrix $A(x)$ with ...
2
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1answer
821 views

Simple example of a moduli space?

This relates to my previous question: Is $\zeta_{\Bbb R^n}$ a space of moduli spaces? Moduli Space I asked two professors about this but they said they didn't know enough about moduli spaces to help....
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0answers
57 views

Dimension of degree $2$ morphisms $\mathbb P^1 \to \mathbb P^1$

I am reading this introduction paper on Gromov-Witten theory, and on page 5, in second paragraph he says: Within this locus of maps, there is a sublocus consisting of those maps which map as $2 : ...
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0answers
77 views

Royal Road to Moduli Spaces

I would like to ask if someone could give me an outline of fundamental works and articles to understand the most important moduli spaces, such as the moduli space of curves, vector bundles, etc... It ...
4
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0answers
193 views

On the functorial point of view in algebraic geometry.

Here's a question I've been thinking about lately. I hope it's not too vague - I apologize in advance if this should be the case. Suppose you want to do algebraic geometry using the $\textit{...
3
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0answers
78 views

Birational equivalence of hypersurfaces

Consider all the smooth hypersurfaces of degree $d$ in $\mathbb P^n$, for $d,n$ general enough. If $X$ and $Y$ are birational equivalent, then I think they are not necessarily isomorphic. I want to ...
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0answers
24 views

Abelian Hitchin's equations

I'm a physicist by trade, and I've recently been working with the dimensional reduction of 4d supersymmetric gauge theories. In particular, the GL-twisted $\mathcal{N}=4$ gauge theory with a simple ...
0
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1answer
110 views

References for Higgs bundles

I would like to ask if someone knows good references for the construction of the moduli space of Higgs bundles from the G.I.T point of view, and the study of this moduli space from the stacky point of ...