# 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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### 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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### 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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### 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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### 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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