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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Family of vector spaces over a scheme

In Example 2.12 of the notes by Victoria Hoskins, the concept of a naive moduli problem is introduced, focusing on vector bundles (locally free sheaves) on a fixed scheme ( X ) up to isomorphism. The ...
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Moduli spaces of surfaces to algebraic stacks

I've been reading through Farb and Margalit's book on the action of the mapping class group on Teichmuller space to get a moduli space. This is a very topological/geometric construction, looking at ...
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Find the center of all circles that touch the $x$-axis and a circle centered at the origin

Given a circle $C$ of radius $1$ centered at the origin, I want to determine the locus of the centers of all circles that touch $C$ and the $x$-axis. This is the red curve in the following Desmos ...
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Elementary question about the definition of moduli space $\mathcal{M}_{g,n}$.

I watched wikipedia page Moduli space, and the definition of $n$-marked moduli space as follow One can also enrich the problem by considering the moduli stack of genus $g$ nodal curves with $n$ ...
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Stability of vector bundles as GIT quotient

I believe that the stability of vector bundles or coherent sheaves (defined as the inequality of 'slopes' of its subsheaves) comes naturally from GIT. However, in any literature I can find, it is ...
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Zariski tangent space to a moduli space

I’m reading the paper 13/2 Ways of Counting Curves by Pandharipande and Thomas. I’m very confused with the following statement on page 8 $\S$ Deformation theory. We return now to the deformation ...
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Euler-Poincaré formula for foliations

Does someone have a nice proof for Proposition 11.14 in Farb&Margalits "Primer to Mapping Class Groups", which states the following: Let $S$ be a closed surface with a singular foliation ...
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Why is this construction of an affine curve not uniformization?

I'm learning the Shimura curve. When I reading the note of Pete L. Clark (SC2-Fuchsian.pdf (uga.edu)), I was stuck on a thinking question. First, there is a theorem (Uniformization Theorem) about ...
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Morphism to moduli stack of curves

I am currently reading up on moduli stack of algebraic curves (https://www.math.uni-bonn.de/~schmitt/ModCurves/Script.pdf). On Page 54 in this script, they claim that giving a morphism $f: S \to \...
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Prestack of coherent sheaves

Alper in his textbook Stacks and moduli defines the prestack of coherent sheaves over a smooth projective curve: Let C be a fixed smooth, connected, and projective curve over an algebraically closed ...
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Doubt about the definition of fine moduli space

I’m studying “Introduction to differentiable stacks” (Grégory Ginot) and I don’t understand a technicality in the (somewhat informal) definition of fine moduli space given at page 12. Basically if we ...
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What is the global complex moduli space for dimensions higher than 1?

I am trying to read the book 'Mirror Symmetry and Algebraic Geometry' by D. Cox and S. Katz. In the book it claims that 'the space of all complex structures on a given manifold $V$ is a well known ...
Hyunbok Wi's user avatar
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Moduli space for $A\sim B$ if $A(D)=B(D)$ for unit ball $D$ and $A,B:\Bbb R\to\Bbb R$ linear

For two linear maps $A, B:\Bbb R^n\to \Bbb R^n$, define the relation: $$ A\sim B \qquad \iff \qquad A(D_n) = B(D_n) \tag1 $$ where $D_n$ denotes the $n$-dimensional ball: $$ D_n=\{x\in\Bbb R^n ~/~ \|x\...
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On Igusa Congruence Groups $\Gamma_g(n,2n)$ and moduli interpretation of $\mathcal{A}_g(n,2n)$

Consider the congruence subgroup $\Gamma_g(n)=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in Sp_{2g}(\mathbb{Z}):\begin{bmatrix}a&b\\c&d\end{bmatrix}\equiv\begin{bmatrix}1_g&0\\0&...
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Definition about moduli space of Riemann Surfaces of genus $g$ with $n$ marked points, $\mathcal{M}_{g,n}$

In the definition of the moduli space of Riemann Surfaces of genus $g$ with $n$ marked points, $\mathcal{M}_{g,n}$, it is asked that $g$ and $n$ satisfy the condition $2-2g-n<0$. I've seen that ...
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The gap between algebraic spaces and DM-stacks

I am following Jarod Alper's course "Introduction to Stacks and Moduli". He gives the following definitions: An algebraic space is a sheaf $X$ on $\mathrm{Sch}_{Et}$ such that there is a ...
Sergey Guminov's user avatar
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What was the difficulty in enumerative geometry problems before physics?

I have read the book 'Enumerative Geometry and String Theory' by Katz, and it left me with some questions. It is outlined in the text how ideas from String theory and TQFT has enriched enumerative ...
Hyunbok Wi's user avatar
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Unclear construction in Hitchins paper on Integrable Systems

In the paper Stable Bundles and Integrable Systems (1987), Hitchin shows that the cotangent bundles of moduli spaces of stable bundles can be viewed as integrable systems. In section 4 of the paper, ...
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What do we know about moduli spaces of sheaves on $\mathbb{P}^n$?

I want to know some examples of moduli schemes of (geometrically) stable sheaves over a higher dimensional base scheme. The simpliest base schemes are the projective spaces $\mathbb{P}^n$ for $n\geq2$....
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Dimension of the moduli space of stable vector bundles

What is the dimension of the moduli space $\mathcal M(r,c_1,c_2)$ of stable rank $r$ vector bundles of fixed chern classes $c_1$ and $c_2$ over an elliptic surface $S$? In the case $r=2$, I read that $...
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Definition of $\operatorname{Lie}(E)$ for an elliptic curve $E$

I'm reading Ehlen's Singular moduli of higher level and special cycles (arXiv link) which uses the following definition of an elliptic curve: Definition 1.2: Let $S$ be a scheme. A proper smooth ...
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Reference for Zariski Main Theorem

Let $\pi: X \to B$ be an elliptic surface with a section $\sigma$. Assume that $\pi, X, B$ are all smooth and that $X,B$ are projective. Let $\mathcal M(r,d)_{X/B}$ be the relative moduli space of ...
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trace and determinant of vector bundles in cohomology

Let $\mathcal{M}$ be a moduli space of stable vector bundles $E$ of fixed invariants over a smooth projective surface $S$,then there is a determinant map $\det: \mathcal{M} \rightarrow \mathbf{Pic}(S)$...
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Homological Algebra issues in showing a moduli functor is representable

Let $h$ be the moduli functor $(\text{Schemes}_{\mathbb{C}})^{op}\to \text{Sets}$ described as follows. On objects $h(X)$ gives us the following: a line bundle $L\to X$ with an injective map of vector ...
daruma's user avatar
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What is Higher Teichmuller Theory?

I am interested in representation of surface group. So, I am studying the book Lectures on Representations of Surface Groups by Francois Labourie. The main goal of representation of surface groups is ...
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GIT stability of plane curve

How do I show that a plane curve of degree d is unstable ( in the GIT sense ) if it has a singular point of multiplicity > 2d/3? This is an exercise from Dolgachev’s book Lectures on invariant ...
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Construction of Deligne-Mumford curve moduli space

The standard construction of $\newcommand{\PP}{\mathbb{P}} \DeclareMathOperator{\Hilb}{Hilb} \DeclareMathOperator{\PGL}{PGL} M_g$ is as follows (At least for stable curves and other technical ...
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Morphism of family of varieties determined on rational points

I am currently studying various Moduli problems and in order to check whether some families have non-trivial automorphisms, I have the strong intuition that the following should hold: Let $k$ be an ...
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representing compactified stack as a global quotient

Let $M$ be a DM stack over a field $k$. Assume I managed to represent it as $[X/G]$ for some scheme $X$ and a group scheme $G$. Does that imply that the DM compactification $\overline{M}$ of $M$ is ...
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Modular forms defined on the modular curve

I am studying the book "Harmonic maass forms and mock modular forms theory and applications" In section 7.5, they deal with p-adic harmonic maass functions. At first, they define the space $...
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Neron polygons and cusps on the modular curve

I am interested in understanding the computation of the number of cusps of $X_1(N)$, which is the modular curve that parameterizes the space of pairs of a generalized elliptic curves $E$ and a point $...
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Rational points on the the reduction mod $p$ of $X_0(N)$

Consider the modular curve $X_0(N)$ over $\mathbb Q$ and for $p\mid N$ consider the reduction modulo $p$ of $X_0(N)$. Let's denote this curve with the symbol $X_p$ (we know that it is a singular curve ...
manifold's user avatar
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Isomorphism in $Coh_{d}(X)/Coh_{d^{'}-1}(X)$

I'm reading "The Geometry of Moduli Space of Sheaves" (Huybrechts,Lehn).On page 26,$Coh_{d}(X)$ is defined as the full subcategory of $Coh(X)$ whose objects are sheaves of dimensions $\le d$....
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Hat Knot Floer Homology with Z coefficients calculation

I would like to ask if there is a reference which carries out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{HFK}(K;\mathbb{Z})$, where the ...
horned-sphere's user avatar
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Hilbert scheme of $\mathbb{P}^2$ is not a product?

Fixing a connected component in the Hilbert scheme of $\mathbb{P}^2$ is the same as fixing the Hilbert polynomial $ax+b$. Taking a primary decomposition of any subscheme in this connected component ...
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Degree of Hodge bundle

Let $H$ be the Hilbert scheme parametrizes subschemes of $\mathbb P^{5g-6}$ with Hilbert polynomial $p(m)=(6g-6)m+(1-g)$ (for example, curves with genus $g$ embedding by the canonical bundle to the ...
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What does "intersect properly" mean?

A corollary in The Geometry of Moduli Space of Sheaves (Huybrechts, Lehn) says: Let $X$ be a normal closed subscheme in $P^{N}$ and $k$ an infinite field. Then there is a dense open subset $U$ of ...
user915579's user avatar
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Why is the definition of the dual sheaf is independent of the ambient space?

I'm reading "The Geometry of Moduli Space of Sheaves" (Huybrechts,Lehn). He gives a new definition of the dual sheaf: $\ $Let E be a coherent sheaf of dimension $d$ ,and let $c=n-d$ be its ...
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Relationship between GIT and coarse moduli spaces

I'm trying to understand how a generic algebraic geometer constructs coarse moduli spaces. I'm familiar with the definition, and how it is usually quite involved to show that a space has the coarse ...
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Are points $(a:b)\in \mathbb{P}^1$ which satisfy $X\subset V(af+bg)$ closed?

Let $X\subset \mathbb{P}^m_k$ be a closed irreducible subvariety, and $f,g\in k[x_0,\dots,x_m]$ be homogenous polynomials of $\deg f=\deg g$. Then we have parametrized hypersurfaces $V(af+bg)\subset \...
Yos's user avatar
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Differences between equivalent definitions of algebraic spaces

I have trouble understanding the differences between the following two equivalent definitions of algebraic spaces. Let $k$ be a field. The first one is An algebraic space is an '{e}tale sheaf $(Sch/S)...
Toney Leung's user avatar
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1 answer
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What does the lattice Λ= Z.z+Z.1 mean?

I'm currently reading The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid and I got stuck (on the fourth page) on the following paragraph: ... the quotient space $\Gamma_1$ \ $\...
lotus57's user avatar
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Coarse moduli space for semi-stable vector bundles on a curve.

Given a curve $C$ of genus $g$ I know how to construct a quasi-projective variety $M_C(r,d)$ as the GIT quotient of a certain Quot scheme $Q=Quot^{r,d}(\mathcal{O}_C(-n)^N)$ by the action of a certain ...
student_du_03's user avatar
3 votes
1 answer
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"Universal family" used despite moduli space not being fine?

This is something that has nagged me for a while, and I think I basically know the issue, but thought I would see what people had to say. In the study of moduli of curves, one will sometimes see the ...
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Hilbert scheme of points on surface is a crepant resolution of symmetric product

If $S$ is a smooth projective surface, it is well-known (I believe first proven by Fogarty?) that the Hilbert scheme of points $\text{Hilb}^{n}(S)$ is a smooth irreducible variety of dimension $2n$. ...
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What does $\mathbb{C}[f]$ mean?

Let $f\colon E \to E$ be an endomorphism of stable bundles over a variety $X$ over $\mathbb{C}$. What does the notation $\mathbb{C}[f]$ mean? (This is supposed to be a field if $f$ is an isomorphism.) ...
fish_monster's user avatar
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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 (\...
nicrot000's user avatar
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9 votes
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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 ...
Cranium Clamp's user avatar
2 votes
0 answers
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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. ...
Yining Chen's user avatar
2 votes
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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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