# 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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### Representability of moduli problem of elliptic curves with complex multiplication

I'd like to know whether the moduli problem for elliptic curves with complex multiplication by a fixed imaginary quadratic number field $K$ (and with suitable level structure to be picked) is ...

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### Teichmüller theory for open surfaces?

I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces?
My motivation basically is that I would like to find out more about the "...

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### Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles

Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...

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### One question about K-moduli space of smooth plane conic curves

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" example 4.5 (2) (b) ADL 19 and l have some confusions.
From Li-Sun's paper "Conical Kähler-...

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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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### 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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### Growth of Betti numbers in moduli spaces of complex stable curves as the number of marked points vary

$\newcommand{\Mgn}{\overline{\mathcal{M}}_{g,n}} \DeclareMathOperator{\nn}{\mathbb{N}} \DeclareMathOperator{\zz}{\mathbb{Z}}$Let $\Mgn$ be the Deligne−Mumford−Knudsen moduli space of stable curves of ...

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### Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?

Following up on this question which received a negative answer, I wonder if something weaker is true.
We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...

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### Symmetric differential forms on moduli space of curves

Do there exist regular symmetric differential forms on $\overline{\mathcal{M}}_{g,n}$ the DM-stack of stable genus $g$ curves with $n$ marked points? By this, I mean nonzero sections $$ \omega \in H^0(...

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### Tangent Space of Moduli of Log-Smooth Curves

We consider an algebraically closed field $\underline{k}$ and all constructions that we will consider are over this field. It is well known that for each relative nodal curve $\underline{f}: \...

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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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### Questions about the Chow varieties

In Lecture 21 of Joe Harris's famous textbook "Algebraic geometry: a first course", he introduced the concept of Chow varieties. In Theorem 21.2, he says that the open Chow variety has ...

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### Inclusion of boundary strata of moduli of curves: induced map on tangent spaces

$\DeclareMathOperator\Ext{Ext}$Let $C \in \bar{\mathcal{M}}_g$ be a nodal curve. It is a classical result that the tangent space of $\bar{\mathcal{M}}_g$ at $C$ is given by
\begin{align*}
T_C \bar{\...

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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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### 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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### Stable curves over non-noetherian schemes

In their seminal paper The irreducibility of the space of curves of a given genus, Deligne and Mumford define a stable curve of genus $g$ over a scheme $S$ to be a flat, proper morphism $X\to S$, all ...

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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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### 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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### The supermoduli space of supertori with odd spin structure and metaplectic group actions

I'm trying to understand the description of the supermoduli space of supertori with odd spin structure as a quotient of the super complex upper half plane $\mathbb{H}^{1|1}$. Such a description ...

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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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### Moduli of smooth curves

Why is the Moduli of smooth curves of a fixed genus not compact/proper?
I know that there is a compactification using stable curves. But is it easy to see that the Moduli of smooth curves is not ...

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### Is multiplication by $d$ on the Jacobian of a nodal curve étale?

Let $k$ be an alg.closed of char$k=0$ and let $A$ be an abelian variety over $k$. This
Lemma on stacks project states that $[d]\colon A\to A$ is étale. In particular, when $A$ is the Jacobian of a ...

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### Is there a *relative* moduli stack of objects functor?

Toen and Vaquie have constructed for any dg category $\mathcal{C}$ a stack $\mathcal{M}_\mathcal{C}$ parametrising objects in $\mathcal{C}$. Its definition is
$$\mathcal{M}_\mathcal{C}(R)\ =\ \text{...

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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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### A question about the book "the geometry and dynamics of magnetic monopoles"

In chapter 2 of the book "The geometry and dynamics of magnetic monopoles", by M.F. Atiyah and N.J. Hitchin (the chapter is called "Geometry of the monopole spaces"), it is written:...

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### How far can one get by counting spaces of solutions this way?

I am quite used to "counting"/computing finite dimensions. For example, one would expect a hypersurface in $\mathbb{C}^3$ to have dimension $3 - 1 = 2$. But it is often the case that the ...

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### When is the morphism from the Hilbert scheme to the moduli scheme of stable sheaves an isomorphism?

Consider over $\mathbb{C}$. Let $(X,\mathcal{O}(1))$ be a smooth projective scheme with an ample polarisation. Let $P(t):=\chi(X,\mathcal{O}(t))$ denote the Hilbert polynomial of $\mathcal{O}_X$. ...

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### Complete curves in $\mathcal{M}_g$ all of whose Jacobians have trivial endomorpism ring

I'm trying to construct complete smooth curves $C$ in $\overline{M}_g$ such that for all points $S \in C \cap \mathcal{M}_g$, its Jacobian $\text{Jac}(S)$ satisfies $\text{End}(\text{Jac}(S)) = \...

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### A couple of questions about the moduli space of annuli with some marked points on the boundary components

I'm trying to work out an answer for my previous question and I'm stuck with the following issue:
In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath ...

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### What is happening on the second step of left mutation?

Let $X$ be a smooth Gushel-Mukai fourfold, whose semi-orthogonal decomposition is given by
$$D^b(X)=\langle\mathcal{K}u(X),\mathcal{O}_X,\mathcal{U}^{\vee}_X,\mathcal{O}_X(H),\mathcal{U}^{\vee}(H)\...

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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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### Examples when algebraic 1-stack = derived enhancement?

Are there any examples where a usual algebraic 1-stack $X$ and the corresponding derived stack enhancement $\mathbb{R}X$ coincide?
Let me take an example from notes of Bertrand Toen, page 41 of https:/...

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### Divisors in moduli spaces of pointed rational curves

I am reading moduli spaces of $n$-pointed rational stable curves denoted by $\overline{M_{0,n}}$. I am not understanding intersection of some divisors as varieties. We know there are forgetful ...

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### Conformal map between flat and hyperbolic torus with a boundary

I am confused because I can define two very different complex structures on the torus with a puncture/boundary.
For my first construction, I can imagine removing a disk from a flat torus, inheriting ...

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### Equivalence between $\bar{\mathcal{M}}_{g,n}$ and ${\mathcal{M}}_{g,n}^{logbas}$

It is a classical result of the theory of the moduli of curves, that the stack $\bar{\mathcal{M}}_{g,n}$ of nodal curves with log-structure coming from the boundary divisor, and ${\mathcal{M}}_{g,n}^{...

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### The precise relationship between (moduli space of) finite-dimensional commutative local $\kappa$-algebras and number theory?

In [BP08] Poonen constructs and studies $\mathfrak{B}_n$, the moduli space of all based $n$-dimensional commutative associative unital $\kappa$-algebras, where $\kappa$ is an algebraically closed ...

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### Finer classification of semistable sheaves

Usually in the moduli space of semistable sheaves, two semistable sheaves correspond to one point if and only if they are S-equvialent, i.e. the graded objects associated to their Jordan-Holder ...

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### Kodaira dimension of spaces of rational curves in hypersurfaces

Let $X\subset\mathbb{P}^n$ be a general hypersurface of degree $d\leq n$, and $\overline{\mathcal{M}}_{0,0}(X,a)$ the Kontsevich space of degree $a$ rational curves in $X$.
Does there exist an ...

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### Discrete subgroups of $\text{Sp}_{4}(\mathbb{Q})$ parameterizing polarized Abelian surfaces plus torsion data

I want to start by considering a familiar congruence subgroup of the integral symplectic group $\text{Sp}_{4}(\mathbb{Z})$. For a positive integer $N$, let $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4}(...

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### On linear schemes

Let $X$ be a smooth projective curve and $T$ is any smooth variety. Let $E$ be a family of vector bundles over $X\times T$ which is flat over $T$. Then there exists a scheme $Y$ over $T$ such that for ...

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### What are étale coverings of the spectrum of a discrete valuation ring?

This question comes when I try the valuative criterion on properness of the moduli space of stable sheaves. Let $X$ be a projective scheme over $\Bbbk$ with an ample line bundle $\mathcal L$. Let $P(t)...

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### Does Albanese construction yield a morphism to moduli of abelian varieties?

Let $M_h$ be the (coarse) moduli space of polarized manifolds with Hilbert function $h$. I would like to know if the albanese $Alb(X)$ of a polarized manifold $X$ gives rise to a morphism $M_h\to A_{g,...

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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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### Definition of modular curve associated to $\Gamma(N)$

For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{...

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### Fundamental group of the moduli space of parabolic bundles with fixed determinant

I am looking for the fundamental group of the moduli space of parabolic bundles with fixed determinant over a smooth projective curve.
I know that the fundamental group of the moduli space of vector ...

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### Cartesian square in the category of Algebraic stacks

Suppose we have a commutative diagram of Artin stacks
$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\...

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### inclusion of moduli spaces induced by morphism between certain universal families

In the recent paper The desingularization of the theta divisor of a cubic threefold as a moduli space, they embed a cubic threefold $X$ into a certain moduli space of stable sheaves. But I do not know ...

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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 ...