A Moduli space is a space in algebraic geometry whose points are geometric objects or isomorphism classes of these kinds of objects.
Questions tagged [moduli-space]
295 questions
27
votes
1 answer
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
- 1,312
18
votes
1 answer
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
- 14,592
18
votes
1 answer
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…
Andres Mejia
- 21,467
17
votes
1 answer
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
- 1,868
17
votes
2 answers
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…
Andrea
- 7,906
14
votes
0 answers
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…
Dan Kneezel
- 351
14
votes
2 answers
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
- 14,592
12
votes
3 answers
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…
mlbaker
- 1,328
- 12
- 25
11
votes
2 answers
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…
Gabriel
- 4,761
11
votes
2 answers
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
- 235
10
votes
1 answer
Moduli space vs. moduli stack of vector bundles
I would like to understand in an intuitive level first and then a technical level also (keeping in mind I am a physicist) the difference between the moduli space of vector bundles and the moduli stack of vector bundles over an algebraic variety or…
Gorbz
- 539
9
votes
2 answers
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
- 5,570
9
votes
1 answer
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…
Cranium Clamp
- 3,423
- 1
- 10
- 30
9
votes
1 answer
Why the existence of automorphism of varieties makes a functor not being a fine moduli space?
Let $F: (Sch) \to (Sets)$ be a functor sends schemes to sets (for example, $F$ sends a scheme $S$ to families of K3 surfaces over $S$ with some fixed polarization). Then it is known that because of the existence of non trivial automorphism (in above…
Li Yutong
- 4,324
8
votes
0 answers
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…
user169903