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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Geometry of the complex Gauge group

This is a pretty naive question: Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$. ...
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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 $...
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Canonical connection on $\mathcal{A}\times X$

Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
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Representations of surface groups

Let $S$ be an orientable and closed surface (compact, without boundary). Its fundamental group admits the following presentation: $$\pi_1(S)=<a_1,...,a_{2g}\, |\, \prod_{i=1}^{i=g}[a_i,a_{g+i}]>$...
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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{...
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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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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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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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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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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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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 ...
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How to see the action of $GL(p)$ on the universal coherent sheaf $\mathcal{U}$ over $X\times Q(\mathcal{O}_X^p/P)$?

In Newstead's book 'Introduction to Moduli Problems and Orbit Spaces' (Chapter 5, Page 110) he says The group $GL(p)$ may be identified with the group of automorphisms of $\mathcal{O}_X^p$; thus $...
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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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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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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 ...
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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{...
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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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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 ...
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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 ...
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To prove that the Elliptic modular function is invariant under the modular transformation

I am not being able to understand that the Elliptic modular function $J(\tau)=\frac{g_2(w_1,w_2)^3}{g_2(w_1,w_2)^3-27g_3(w_1,w_2)^2}$ is invariant under the modular transformation $\tau\mapsto \frac{a\...
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Moduli Space of Elliptic curves

I am trying to see that: The moduli space of Riemann surfaces of genus 1 with one marked point is $\mathbb{H}/PSL(2,\mathbb{Z})$. I know the facts that $PSL(2,\mathbb{Z})$ acts on the upper half ...
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Why are universal families useful?

I recently started to read about moduli spaces. The best one can probably hope for, seems to be a fine moduli space and that is a moduli space that carries a universal family. In other words, the ...
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Understanding the local structure at a point on the moduli of semistable bundles of rank 2 of fixed determinant over a curve

First, let me describe the premise. Let $X$ be a smooth projective curve of genus greater than 2 over $\mathbb{C}$, let $x,y\in X$ two closed points, and let $M_{y-x}$ denote the moduli space of S-...
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What is the Moduli space and what are the elements of it?

What is the Moduli space and what are the elements of it? So far I know that Moduli space is a vector space with each vector a "equivalence class of objects". Is it true that each member of a Moduli ...
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Quot-like scheme for torsion sheaves

I am wondering if, as the Quot schemes parametrizes flat (quotients of) sheaves over schemes, there is anything similar for torsion sheaves. In first approximation, if $I$ is a sheaf of ideals over a ...
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What is the multidegree of a curve $C \subset \mathbb{P}^n \times \mathbb{P}^m$?

What is the multidegree of a curve $C \hookrightarrow \mathbb{P}^n \times \mathbb{P}^m$? I'm reading Notes on stable maps and quantum cohomology by W. Fulton and R. Pandharipande, and on page 14, ...
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Automorphisms of stable curves.

I am a bit confused by what Harris and Morrison write about the finiteness condition for stable curves in Moduli of Curves. First the defintions: Definition (2.12) A stable curve is a complete ...
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space of projective plane curves of degree d

This is basically from Hartshorne Exercise I.5.13 where he writes there is a correspondence from the set of plane projective curves of degree $d$ to points in a projective space $\mathbb P^N$ where $N=...
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Why is the space of plane quadrilaterals with area center of mass = edges center of mass = vertices center of mass 2-dimensional?

A naiv geometric question. We consider shapes of triangles and plane quadrilaterals (i.e. these objects up to translation, rotation and scaling). The shape of triangles ($T$) is 2-dimensional (...
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Automorphisms and moduli problems

So I've been trying to understand why in general automorphisms stop a moduli problem being representable. I've seen the answer to Killing the automorphisms to make a functor representable To see ...
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Not all curves of genus greater or equal to 3 are hyperelliptic, and their dimension

I'm supposed to show that for all genuses $g\geq3$, not all curves are hyperelliptic. Let me preface this by saying that I am not taking a what I think probably is a typical algebraic geometry course. ...
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Families of genus zero curves vs Family of deformations of $\mathbb{P}_k^1$.

Suppose I am looking to classify isomorphism classes of connected, compact, projective curves of genus zero. We usually start this moduli problem by considering a scheme $S$ and families of genus ...
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118 views

Every quadratic polynomial is conjugate to one of the polynomial $z^2+t$

Every quadratic polynomial is conjugate to one of the polynomial $z^2+t$ where conjugation means conjugation by a linear fractional transformation $\alpha(z)=\frac{az+b}{cz+d}$ and $f^{\alpha}={\...
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Hilbert scheme of n points and moduli space of ideal sheaves

Let $X$ be a smooth projective variety. I have seen people identify the Hilbert scheme of $n$ points on $X$ with the moduli space of stable sheaves with Chern character $(1,0,....,-n)$ (say on $\...
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Moduli space of vector bundles open in moduli of sheaves

Let $X$ be a smooth projective variety with polarization $\mathcal{O}_X(1)$. For fixed Chern class $r,c_1,c_2....$, do we have an OPEN embedding $$M^{ss}(X,r,c_1...)_{vec}\subset M^{ss}(X,r,c_1...)$$ ...
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First order deformation isomorphic to the tangent space of moduli?

Let $\mathcal M$ be the moduli space of all smooth hypersurface of degree $d$ and dimension $n$. Is it true that $$T_{[X]} \mathcal M \cong Def(X):=H^1(X,T_X)$$ holds for any such hypersurface $X$? ...
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natural fiber evaluation map of stable maps

I want to understand page 31 in FP-notes by Fulton and Pandharipande. Suppose that $K=D(A;B;D_1;D_2)$ be a boundry divisor of $\overline{M}_{0,n}(X,\beta)$ and let $\overline{M}_A=$ $\overline{M}_{0,...
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Moduli space of stable curves of genus $g$ does not admit a universal family

I am trying to understand why the (coarse) moduli space $\overline{M}_g$ of stable curves of genus $g$ does not admit a universal family. I am following the proof in p.267 of this book. A key step is ...
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108 views

Is the Teichmuller space of a surface always a contractible CW-complex?

I'm trying to prove that the classifying space of the mapping class group of a surface $S$ with genus $g$ and $n$ boundary components is homotopy equivalent to the moduli space $\mathcal{M}_{g,n}$. ...
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Size and shape moduli of Calabi-Yau manifolds

Let's say that Calabi-Yau manifolds are compact Kähler manifolds with a trivial canonical bundle. We know that $H^2(X,\mathcal T_X) = H^1(X,\Omega^2_X)$ is the tangent space to the deformation functor ...
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Monodromy of the family of hypersurfaces on moduli space

Let $\bar{\mathfrak X}\to \mathbb P^N$ be the universal family of hypersurfaces in $\mathbb P^{n+1}$ of degree $d$ and $\mathfrak X \to U$ ($U\subset \mathbb P^N$) be the sub-family of smooth ...
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Question on local coordinates in the moduli space $\mathcal{H}_g(1)$

Fix $g\ge 2$ and denote by $\mathcal{H}_g(1)$ the moduli space of holomorphic 1-forms on a closed Riemann surface of genus $g$ with one zero of order $2g-2$. Given any point $\omega\in \mathcal{H}_g(1)...
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Why is the Hilbert scheme $ \operatorname{Hilb} ( n )_S $ a subscheme of a Grassmannian to determine?

In the following link of Wikipedia : https://en.wikipedia.org/wiki/Hilbert_scheme , there is a short sentence in the paragraph : Construction, which says : Grothendieck constructed the Hilbert ...
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Conditions for proving infinite codimension?

I am working with moduli spaces of curves and I am not used to work with infinite dimensional spaces. Then I was wondering if there are sufficient explicit conditions to stablish that certain subspace ...
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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 ...
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Is there a moduli space $\mathcal{M}$ of the closed, $n$-dimensional, simply connected manifolds?

Is there a moduli space $\mathcal{M}$ of the closed, $n$-dimensional, simply connected manifolds with the following properties: (i) Two abstract simply connected closed manifolds $M, N\in\mathcal{...
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Any representable moduli problem of elliptic curves is rigid

EDIT: As I now tried to provide an answer to this problem, I would appreciate any proof verification of my reasoning. Thank you very much! In Katz and Mazur's book (available here, page 109, page 60 ...
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Equivalent definitions of $\Gamma _1(N)$-structures on elliptic curves

In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), two equivalent definitions of a $\Gamma _1(N)$-structure (point of exact order $N$) are given on page 99 (page 55 in ...
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on the maps to moduli spaces

Let $M$ be a coarse moduli scheme for some moduli functor $\mathcal{F}$( from the category of schemes to the category of sets). Let $V$ be a scheme such that for each point of $v \in V$ we can ...
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Does $ H^1 (X , \mathcal{O}_X^{ \ * }) $ represent the following moduli functor?

We know that Picard's group of a compact manifold $X$, denoted $\mathrm{Pic} (X)$ is the group of isomorphism classes of holomorphic line bundles, with the tensor product operation. We also know that ...