Questions tagged [moduli-spaces]
Given a concrete category C, with objects denoted Obj(C), and an equivalence relation ~ on Obj(C) given by morphisms in C. The moduli set for Obj(C) is the set of equivalence classes with respect to ~; denoted Iso(C). When Iso(C) is an object in the category Top, then the moduli set is called a moduli space.
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Which curves cut the Hyperelliptic locus?
Consider the moduli space $\mathcal{A}_{g,n}$ of abelian varieties with some level $n \geq 3$ structure. For simplicity, we just denote it by $\mathcal{A}_{g}$ and drop $n$. Denote the locus of ...
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223
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Possible slope for a modular form
Let $f\colon \mathbb{H}_g \to \mathbb{C}$ be a Siegel modular form of weight $k$ with respect to $\Gamma_g$.
Then, $f$ admits a Fourier expansion $f(Z) = \sum_T a(T) \exp(i\pi \mathop{tr} TZ)$, where $...
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217
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Level n-structure as defined by Mumford in GIT
In Mumford's GIT, the definition of level $n$ structure ($n \geq 2)$ is $2g$ sections $\{\sigma_1, \dots, \sigma_{2g}\} : S \rightarrow A$ such that two conditions hold: (i) For geometric points the ...
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303
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Moduli space of points of fixed order N on elliptic curves
Let us consider the moduli space $Y_1(N)$, parametrizing an elliptic curve, together with a choice of a point of N-torsion on it. Over the complex numbers, it is easy to see that this moduli space is ...
2
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271
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Is there an algebraic analogue of the degeneration of riemann surfaces in M_g
Degeneration of certain functions such as theta functions or Green's functions in the moduli space $\overline{\mathcal{M}_g}$ of stable curves of genus $g$ has been studied quite alot. The idea is to ...
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252
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Projectivized Normal Cone to Satake Compactification
Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties over $\mathbb{C}$.
There exists a compactification, the Satake compactification, which is minimal and has the ...
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684
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Vector bundles on some non-projective surfaces
Let $X$ be a smooth projective curve over a field $k$ and let $L$ be a line bundle on $X$.
I will denote by $S$ the total space of $L$ -- this is a smooth surface over $k$ containing
$X$ (as the zero ...
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177
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How to characterize good "models" of a category
Let ${\bf Cat}$ denote the category of small categories. Recall that for a category $\mathcal{C}$ and a functor $F\colon\mathcal{C}\to{\bf Cat}$, the Grothendieck construction of $F$, which I'll ...
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82
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Hopf algebra from Chow rings of Hilbert schemes of smooth surface
Let $X$ be a smooth projective surface. As its Hilbert schemes of points are resolutions of the symmetric powers, the addition map $S^nX \times S^mX \rightarrow S^{n+m}X$ lifts rational map $X^{[n]} \...
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87
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About the relationship between Cayley-Chow families and well-defined family of proper cycles
I'm studying Chow varieties introduced in Chapter I.3-4 of "Rational curves on algebraic varieties" [Kol96] by János Kollár and also very interested in the "open" Chow variety ...
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Questions about the Chow varieties, II
This question is closely related to my previous question.
Recently, I find another version of the open Chow variety in János Kollár's book Families of varieties of general type. I guess that (3.5) and ...
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Moduli stack of l-adic sheaves?
Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies étale $G$-torsors.
Let $\ell$ be a prime ...
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80
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What is the meaning of universal family of Fulton Macpherson configuration space?
Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces"
In this paper, the process ...
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132
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Locus where a family of cycles is rationally trivial is closed?
Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...
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130
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Chainsaw quiver variety and parabolic bundle
How can we relate chainsaw quiver varieties with ADE type Nakajima quiver varieties?
We know that we can obtain ADE type quiver varieties (instantons over ALE spaces) by taking $\Gamma$ equivariant ...
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What is bad when stabilizers are non-reductive in moduli stacks?
Here is J. Alper's definition of good moduli spaces.
Consider in characteristic zero. Then we see that the classifying stack of any non-reductive group $H$ does not have a good moduli space. In ...
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Homology groups of moduli of parabolic bundles with fixed determinant
I am looking for the Homology groups of the moduli space of stable parabolic bundles over a smooth projective curve with fixed determinant.
In particular, what is the second homology group of such ...
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Local Chart for Teichmuller Space as A Manifold
Let $(R,p_1,…,p_n)$ be a Compact Riemann surface of genus $g$ with $n$ marked points. Its deformation space is $H^1(R, K_R^{-1}\otimes\mathcal{O}(-p_1),…,\mathcal{O}(-p_n))$. From Riemann-Roch theorem,...
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What is the semistable reduction for sheaves?
Let $\Bbbk$ be an algebraically closed field with characteristic zero. Let $X$ be a projective scheme over $\Bbbk$ and let $L$ be an ample invertible $\mathcal{O}_X$-module. Fix a Hilbert polynomial $...
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85
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Intersection of two quadrics as moduli space
Let $Y:=Q_1\cap Q_2\subset\mathbb{P}^{n-1}$ be smooth complete intersection of two quadrics. If $n$ is even, then it admits a semi-orthogonal decomposition:
$$D^b(Y)=\langle D^b(C),\mathcal{O}_Y,\...
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Moduli space of abelian surfaces
Let $K$ be a finite field with algebraic closure $\overline{K}$. The $j$-invariant gives a bijection between the the affine line $\mathbb{A}_K^1$ and $\overline{K}$-isomorphism classes of elliptic ...
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Moduli of morphisms between varieties
Let $\mathcal{M}, \mathcal{N}$ be two "well-behaved" (i.e. representable by an algebraic stack parametrising flat, proper, surjective, finitely presented morphisms with geometric fibres ...
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71
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Action of involution on instanton bundle
Let $Y$ be a quartic double solid and $E$ be an rank two instanton bundle on $Y$. By Serre's correspondence, it is not hard to show that $E$ fits into the following short exact sequence $0\rightarrow\...
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Curve without infinitesimal automorphism has no deformation with automorphism
$\DeclareMathOperator\Spec{Spec}$From studying the classical literature on deformation theory, I have the impression that if the identity morphism of a curve $C$ over an algebraically closed field $k$ ...
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73
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What is the functor of points of the moduli scheme of stable sheaves?
Let $\Bbbk$ be an algebraic closed field of characteristic zero. Let $(\mathrm{Sch}/\Bbbk$ denote the category of locally Noetherian schemes. Let $B$ be a projective scheme over $\Bbbk$. Let $L$ be an ...
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A question about Galois group action on sheaves, descent theory and $\mu$-semistable sheaves
Let $f : Y \to X$ be a finite morphism of degree $d$ of normal projective varieties over $k$ of dimension $n$. In Lehn and Huybrechts' book "The Geometry of Moduli Space of Sheaves", there ...
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$\mu$-polystable locally free sheaf
In Huybrechts and Lehn's book "The Geometry of Moduli Space of Sheaves", a sheaf $E \in Ob(Coh_{d,d-1}(X))$ is polystable if $E \cong \bigoplus E_{i}$ in $Coh_{d,d-1}(X)$, where the
sheaves $...
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422
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Hilbert scheme of divisors in smooth projective varieties
Let $X$ be a smooth projective variety and $L$ be a line bundle with $H^0(X,L)\neq 0$. Let $D\in |L|$ and $p(t)$ be the Hilbert polynomial of $D$. Assume that any effective divisor $D'\subset X$ with ...
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141
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Moduli interpretation for integral models of PEL Shimura variety at parahoric level?
Kottwitz has built canonical integral models for a large family of PEL Shimura varieties, associated to a certain reductive group $G$ over $\mathbb Q$, when the structure level has the form $K = K_pK^...
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146
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Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
I need the reference to a detailed proof the following fact.
Let $g\geq 2$ be an integer. Let $\mathcal M^{ss}_g: Sch\rightarrow Groupoids$ be the prestack which sends a scheme $T$ to the groupoid of ...
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106
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Does there exist other known pair of Fano threefolds/fourfolds with residue categories being K3/Enriques?
Let $Y$ be a Gushel-Mukai threefold, we can either consider an ordinary Gushel-Mukai fourfold $X$ containing $Y$ as a hyperplane section, or we consider a special Gushel-Mukai fourfold $X'$ as double ...
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Extension of a section of a line bundle on a family of curves to the central fibre
I will fix notation: $\Delta = \mathrm{Spec} R$ denotes a discrete valuation ring and $\Delta^*=\mathrm{Spec} K$ for $K=\mathrm{Frac}(R)$. Suppose we are given a curve $\pi:C\to \Delta$ and a line ...
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103
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Computation of semistable model of curve over DVR of mixed characteristic
given a stable curve $C \in \overline{\mathcal{M}_{g,n}}(K)$ over a field $K$ and an DVR $R$ with fraction field $K$ and algebraically closed special point, is there a general technique to compute a ...
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195
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Ex 1.1c Hartshorne Deformation Theory: Is this family flat?
This comes from my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ...
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0
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70
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Gluing of hybrid trajectories in Floer homology
In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
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76
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Banach manifold structure on the moduli space of hybrid trajectories
I am reading the paper "On the Floer homology of cotangent bundles", (arXiv link) , by Abbondandolo and Schwarz and in page $35$ to define the isomorphism between the Morse complex and the ...
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136
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Koszul cohomology and nodal curves
In M. Aprodu, G. Farkas - Koszul Cohomology and Applications to Moduli, arXiv:0811.3117 [math.AG], proof of Theorem(s) 4.5 (and 4.12), the authors constructed a semistable nodal curve $C^{\prime}$ of ...
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84
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Moduli of semistable sheaves as fiber bundle
Let $X$ be a smooth projective variety over a field of char 0, and $F$ a stable $\mathscr{O}_X$-module of rank $r$ and determinant bundle $Q$. There is a natural morphism $\operatorname{det}:M\to \...
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146
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Properness of Hom-schemes for finite group schemes
In SGA 3 XI proposition 3.12 (b) (https://webusers.imj-prg.fr/~patrick.polo/SGA3/Expo11.pdf)
It is shown that:
If $G$ is an affine group scheme over a base $S$ and $H$ is a finite group scheme over $S$...
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85
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semi-orthogonal decomposition of Fano fourfold associated to threefold
Let $Y$ be Gushel-Mukai threefold and $X$ a Gushel-Mukai fourfold containing $Y$ as its hyperplane section, the semi-orthogonal decomposition of $X$ and $Y$ are both known. Also, for cubic fourfold ...
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Local description of the universal family $\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}$
I would like to get an understanding of the notion of geometric fibers of the universal family:
$$\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}.$$
In fact Knudsen show ...
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139
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Codimension of the complement of the stable locus
Let $\mathcal{M}$ denote the moduli space of semistable vector bundles of fixed rank and degree over a compact Riemann surface $X$. Let $\mathcal{M}^s \subset \mathcal{M}$ be the moduli of stable ...
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111
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On stability of coherent sheaves over a quasiprojective variety
Let $X$ be a smooth projective variety of dimension $n$ and $L$ be an ample line bundle on $X$. For any coherent sheaf $E$, one can define the first Chern class $C_1(E)$ and degree of $E$ to be $C_1(E)...
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257
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Mirzakhani's work and surfaces with marked points on the boundary
Mirzakhani proved identities for the lengths of geodesic curves on Riemann surfaces of genus $g$ and with $n$ boundary components. She used these to provide an integration scheme over the ...
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131
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Bubbling of disks when proving compactness properties in Lagrangian floer cohomology
When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the ...
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The uniqueness of some semistable torsion free sheaves on Fano threefold
Let $X$ be a prime Fano threefold of index one and even genus $g\geq 6$, one can show that the moduli space of torsion free semistable sheaves $M(2,1,m_g)$ with $m_g=\left \lceil{\frac{g+2}{2}}\right \...
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102
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Tangent space of moduli of stable vector bundles
I'm new to this area, so it may very well be possible that I may be missing something easy here.
Let $E$ be a stable complex vector bundle over $X$ of degree $d$ and rank $n$. Then the moduli space $\...
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125
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Schur's lemma for sheaves with different reduced Hilbert polynomials
Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement:
Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
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151
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Moduli space of genus $g$ curves ${\mathcal{M}_g}$ irreducible by 'Monodromy argument'
I'm reading this post by Charles Siegel on Monodromy Representations
and there is a short remark on the proof of irreducibility of moduli space of genus $g$ curves ${\mathcal{M}_g}$ :
Just look at ${...
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0
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78
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A question about Hitchin discriminant
Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be moduli space of Higgs bundle of rank $r\geq2$ and degree $d$ on $X$, $W=\bigoplus_{i=2}^{r}H^{0}(X,K_{X}^{\otimes i})$, and $H\...