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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 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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deformations of $p$-divisible groups

I've been wanting to learn more about the deformation theory of $p$-divisible groups and was looking for some references. I have looked into stuff like the Serre-Tate theorem with Katz's Serre-Tate ...
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A question about Gieseker compactification

Let $\ell_{\infty} \subset \Bbb P^2$ be a fixed line, and $G = GL_r$. Let $\mathcal U^a_G$ be the Gieseker partial compactification of the moduli space $Bun^a_G(\Bbb A^2)$. The space $Bun^a_G(\Bbb A^2)...
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Base change and family of stable maps

Suppose that family F of stable maps given by maps $f:C \to S,\mu:C \to P^r$ and sections $\rho_i:S \to C$ Suppose that $\Sigma(F)$ be union of all one dimensional components of locus of nodes in ...
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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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Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $-2\pi i \Omega$...
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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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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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Vector bundles are reps of $\Gamma_\mathbb{R}$

It's well-known that every flat holomorphic vector bundle $V$ of rank $n$ on complex manifold $X$ comes from a representation $$\rho\ : \ \pi_1(X) \ \longrightarrow \ \text{GL}(n,\mathbb{C})$$ by ...
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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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Fine moduli space of rigid stable families to projective space

In FP-notes. in Theorem 3, pages 16-17, in order to prove that a functor of rigid stable families (in genus zero) has fine moduli space, they defined a $H$-balanced morphism and they show that we have ...
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Flatness of Morphism between Schemes

Let $S$ be a smooth projective surface (by surface here I mean a $2$-dimensional, proper $k$-scheme) and $\mathcal{L}$ be an invertible sheaf. Then we can consider the projectivation of $\mathcal{L}$...
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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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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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How to calculate $ H^{ \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) $?

Can you explain to me please, how to calculate the rational cohomology algebra $$ H^{ \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) = \displaystyle \bigoplus_{k=0}^{+ \infty} H^{ ...
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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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Is it always possible to construct a rigid vector bundles with given chern character?

Let $X$ be a del Pezzo surface over $\mathbb{C}$. Given an integer $a\geq 1$ and a divisor $C$ on a del Pezzo surface $X$, is it always possible to construct a rigid vector bundle $\mathcal{E}$ (i.e $\...
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Moduli space for line segments through the origin?

The real projective line $P^{1}(\mathbb{R})$ is the space of all rays though the origin of $\mathbb{R}^{2}$. If I instead consider line segments, rather than infinite straight lines, and ask for a ...
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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 ...
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Is the concept of vector bundle is a special case of the notion of family of algebraic varieties?

If we start from the definition of the notion of families of varieties ( or algebraic ( projective ) varieties more precisely ), as proposed by Joe Harris in his book: Algebraic geometry: A first ...
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Does every homeomorphism of a Riemann surface have an isotopic biholomorphism?

I'm trying to understand the definition of the moduli space of genus $g$ Riemann surfaces as the quotient space of the Teichmuller space by the action of the mapping class group. I think I have an ...
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For which CM points $\tau$ is $\gamma(\tau)$ also a CM point?

Let $\tau$ be a CM point in the upper half plane $\mathcal{H}$ - that is, an element of $\mathcal{O}_K$ for an imaginary quadratic extension $K/\mathbb{Q}$ that lies in $\mathcal{H}$ after choosing an ...
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General twisted cubics

In Joe Harris's text book entitled : Algebraic geometry, a first course, page : $ 55 $, we find the following paragraph which i don't really understand : ... As another example, consider again ...
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Why is any coarse moduli space for hypersurfaces is a categorical quotient?

Hypersurfaces of degree $ d $ in $ \mathbb{P}^n $ are parametrized by points in the space $ k[x_0 , \dots , x_n ]_d \backslash \{0\} $ of non-zéro degree $d$ homogeneous polynomials in $n+1$ variables....
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Characterization of Kahler-Einstein manifolds

How can we characterize Kähler-Einstein manifolds by using cohomological method? More precisely How can we charactrize Fano Kähler-Einstein manifolds by using cohomological method? Any ...
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Canonical metric when $-K_X$ is nef?

Let $X$ be a smooth projective variery. Let $-K_X$ be nef , then which type of Canonical metric In the sense of Einstein type metric is suitable for it. In fact when $K_X$ or $-K_X$ is ample WE know ...
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Concavity of moduli space of Kahler-Einstein manifolds

Conjecture of Zhiqin Lu : Calabi-Yau moduli space is a concave manifold?. Definition: By concavity, we mean that there exists an exhaustion function on the manifold such that at each point the (...
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Action of quotient of congruence subgroups on moduli space

Let $\Gamma(N) \leq \Gamma_{1}(N) \leq \Gamma_{0}(N)$ be the usual congruence subgroups of the modular group $SL_{2}(\mathbb{Z})$, with all containments normal. We have, e.g., the quotient group $\...
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Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface

Assume that $\mathcal{R}$ is a hyperbolic surface with $m$ geodesic boundary components and $n$ punctures. If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is ...
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Projection $\pi:\overline{\mathcal{M}}_{0,5}\to\overline{\mathcal{M}}_{0,4}$.

I am trying to show that the the forgetful map $\pi:\overline{\mathcal{M}}_{0,5}\to\overline{\mathcal{M}}_{0,4}$ is complex analytic. The description that I have of this map is purely geometrical: it ...
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Hyperbolic Metric on a Riemann Surface

From uniformization theorem, it is known that every conformal class of metrics on a Riemann surface contains a unique hyperbolic metric. For a genus-$g$ Riemann surface with $n$ punctures, the ...
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Why is $ \mathbb{N} $ a fine moduli space for the moduli problem of finite sets up to bijection?

How to establish rigourously that $ \mathbb{N} $ is a fine moduli space for the moduli problem of finite sets up to bijection ?. Thanks in advance for your help.