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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What is a moduli space for a differential geometer?

A moduli space is a set that parametrizes objects with a fixed property and that is endowed with a particular structure. This should be an intuitive and general definition of what a moduli space is. Now, in the contest of algebraic geometry, we…
User3773
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The Moduli Stack of Elliptic curves - What is it?

I have often heard the words "Moduli Stack of Elliptic Curves", but I have nowhere found a from-scratch definition of this object. I do understand the motivation: There are cusps in the moduli space that produce singularities. For me, a stack is a…
Kofi
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What are D-branes (in a topological field theory)?

In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as habitats for the ends of open strings and can be…
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Hilbert polynomial and Chern classes

Motivation: moduli spaces of semistable sheaves. Let $(X,\mathcal O_X(1))$ be a smooth projective variety over a field $k=\overline k$. When one defines the moduli functor $\mathcal M_P:\textrm{Sch}_k\to \textrm{Sets}$ of semistable sheaves, one…
Brenin
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Concrete Problems that can be solved by appealing to a Moduli Space

I have always enjoyed the idea of creating "parameter spaces" or "moduli spaces," but it is only recently that I have seen very concrete applications of studying the moduli space. Because of how pervasive this theory is, I was hoping that Notable…
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Good books/expository papers in moduli theory

I have been studying mathematics for 4 years and I know schemes (I studied chapters II, III and IV of Hartshorne). I would like to learn some moduli theory, especially moduli of curves. I began reading "Harris, Morrison - Moduli of curves", but I…
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The coarse moduli space of a Deligne-Mumford stack

This question is just a "definition request". Question. What is the "the coarse moduli space" of a Deligne-Mumford stack? I know (the basics of) DM stacks, but how are their moduli spaces defined? What is their link with moduli? Are they…
Brenin
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Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The Jacobian of $K$ is a product of three elliptic curves…
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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 the underlying topological spaces? I know that…
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Moduli space of isogeny classes of elliptic curves

The modular curve $Y(1)$ classifies isomorphism classes of elliptic curves, namely its $K$-points for any field $\mathbb Q\subseteq K\subseteq \mathbb C$ correspond via the $j$-invariant to $\mathbb C$-isomorphism classes of elliptic curves defined…
Ferra
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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 would spoonfeed me the math a little. I am…
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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{Pic}(\mathbb A^1) = \{\mathcal O_{\mathbb…
JoS
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Hilbert Schemes and Moduli Spaces

The formal definition of a fine moduli space involves a representable functor(the detailed definition is not quoted here for simplicity). According to it, in the most general case, just any scheme can be a moduli space, because it represents its own…
Ashson
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How is moduli of curves relevant in physics?

From: Moduli space we see that moduli of curves is a very algebro-geometric topic. It is easy to understand its relevance and importance in algebraic geometry. But the mind boggles when we try to imagine how on earth such a topic from pure and…
Wilkinson
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Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with essential topology, general relativity, Hamilton–Jacobi…
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