# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 ...
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 ...