# 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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### A Subfunctor of Quot-functor compatible with pullbacks

Let $X$ be a smooth projective irreducible algebraic curve over field $k$. For $d,r,k,m >0$ the representable Quot scheme $\mathcal {Quot}_X^{r,d}(\mathcal{O}_X(m)^k)$ is given for
any test scheme $...

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198 views

### Mapping Class Group and Triangulations

I am a physicist who's getting started with Mapping Class Group for Riemann surfaces, pants decompositions and triangulations so I apologise in advance if the following is a stupid question/wrong.
I ...

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59 views

### 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\...

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### Quadrics tangent to lines

I think that the following must be a basic question in enumerative geometry.
Take a line $L\subset\mathbb{P}^3$. The quadric surfaces in $\mathbb{P}^3$ that are tangent to $L$ are parametrized by a ...

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### The dimension of parameter space of unstable Higgs bundle

Let $X$:smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$, $\mathcal{M}(r,d)$:moduli space of stable Higgs bundles of rank $r\geq 2$ and degree $d$ on $X$, and $N$:moduli space of stable ...

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170 views

### Moduli space of annuli with marked points satisfying some additional symmetries

Let us consider the space of configurations $\overline{\mathcal{M}}_{0,2,1,(1,1)}$ of an annulus with a marked point on the interior boundary component (let's call it "out") a marked point ...

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### Symplectic form on the space of unitary connections $\mathcal{A}(E)$

Let $E\rightarrow X$ be a Hermitian vector bundle over a (Kahler) manifold $X$. The space of unitary connection $\mathcal{A}(E)$ is an affine space modelled over $\Omega^1(X,u(E))$ and is endowed with ...

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### Upper bounds for the degree of Chow varieties

Given $n, k, d$, let $\mathrm{Chow}(n, k, d)$ be the Chow variety parameterizing algebraic cycles of pure dimension $k$ and degree $d$ in $\mathbb{P}^n$. It is a projective subvariety of $\mathbb{P}(H^...

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### global algebraic functions $\Gamma(T^{*}M)$ on the cotangent bundle of moduli space

Let $X\colon$ smooth projective curve,
$\mathcal{M}\colon $ moduli space of semistable higgs bundle of rank $r$ and with fixed determinat $\xi$, and
$H\colon \mathcal{M}\rightarrow W=\oplus_{i=2}^{r} ...

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201 views

### Moduli space of genus 1 curves with a degree n divisors

I am sure this is well known, but I don't know what to search for:
Consider $M_{1,n}$, the moduli space of genus 1 curves with $n$ marked points. The symmetric group on $n$ letters acts on this space ...

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129 views

### Smoothness of moduli spaces of stable maps

If $X$ is a projective variety the moduli space of stable maps $\overline{M}_{0,0}(X,\beta)$ is a normal variety with finite quotient singularities.
Can the pairs $(X,\beta)$ such that $\overline{M}_{...

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53 views

### Nakamura graphs and moduli space cellular decomposition

I have recently started studying the cell decomposition of moduli spaces. Among the papers I read, I studied this paper, but there is something I do not understand and I can't find the answer on my ...

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233 views

### On logarithmic schemes

I have two questions on logarithmic schemes
Can we explicitly construct a chart for any coherent logarithmic scheme? By definition of coherence it must have a chart but given a coherent sheaf of ...

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### Log point and monoid structure

My question is related to the definition of the standard Log Point.
It is defined in the following way: Let $k$ denote any field of $Char ~~0$ and $\mathbb N$ be the monoid of Integers. Then the map $\...

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### The openness the set of $s\in \bigoplus H^0(C,K_{C}^{\otimes i})$ for which the spectral curve is irreducible and reduced

Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...

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136 views

### Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold

Let $X$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $X\rightarrow\mathrm{Gr}(2,5):=G$. Consider the normal bundle $\mathcal{N}_{X|G}$, how to compute cohomology of this ...

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### Global algebraic function over the moduli space of semistable higgs bundles $\mathcal{M}$

Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be the moduli space of semistable higgs bundles, and $h: \mathcal{M}\rightarrow W=\bigoplus H^0(X,K_X^{\otimes{i}})$ be the ...

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140 views

### Cotangent bundle of moduli space of stable bundles

I think this is a basic and dumb question.
Let $X$ be a smooth projective curve over $\mathbb{C}$, $M$ be a moduli space of stable bundles and $\mathcal{M}$ be a moduli space of semistable higgs ...

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196 views

### Fibers of Hitchin fibration are equidimensional

Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g\ge 3$, $M$ be a moduli space of stable vector bundles on $X$ of rank $n\ge 2$ and degree $d$, $\mathcal{M}$ be a moduli space of ...

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### Explicit check of the invariance of the Weyl-Petersson form

Using Fenchel-Nielsen coordinates, the Weyl-Petersson metric can be written as
$\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$
where $i$ is an index labelling the curves of a pants decomposition ...

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69 views

### Polarization of Prym varieites

I'm trying to understand polarization and rational Hodge structure of spectral curves and Prym varieties.
Excuse me that this is similar to my previous question.
I want to prove the following,
Let $X$...

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### The compactified Jacobian is birational to a $\mathbb{P}^1$-fibration over the Jacobian of normalization

Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\...

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### Correspondence between Riemannian metrics and Euclidean embeddings

Given a sufficiently smooth manifold M,
a Riemannian metric on M induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely)
an embedding of M into ...

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### Fiber of the Hitchin map

Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...

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153 views

### Pants decomposition and moduli space of $\Sigma_g$ for $g>1$

By Section 8.3.1 of the book: A primer on mapping class groups by Farb and Margalit, a pair of pants is a compact surface of genus 0 with three boundary components. Let $S$ be a compact surface with $\...

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141 views

### Kodaira-Spencer map in logarithmic geometry

Can anyone provide a reference for the Kodaira-Spencer map in the logarithmic geometry setting?

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### Does anyone know if it's possible to construct Moduli space of J holomorphic curves using Holder spaces?

let Y be a contact (3) manifold and X be its symplectization. let's say the Reeb dynamics is at least Morse Bott. let $u: \Sigma \rightarrow X$ be a $J$ holomorphic curve. I know the usual model for a ...

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### Methods to compute the Kodaira dimension of moduli spaces

It is known that the moduli space $\bar{M_g}$ of genus $g$ stable curves over $\mathbb C$ is of general type for $g \geq 24$ with Kodaira dimension $3g-3=\dim \bar{M_g}$.
The idea is that one can ...

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### Geometry of moduli problem in practice: how to check it is connected / irreducible / normal / reduced / locally complete interesection…?

Moduli spaces are very common and useful in the world of algebraic geometry. From the point view of functors, one can already check many geoemtric properties of it. I like examples, and you can assume ...

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### Moduli-space interpretation of a morphism of unitary Shimura varieties

Let $G$ be the quasi-split unitary similitude group $GU(2, 1)$, for some choice of imaginary quadratic field $E$; and let $T = GU(1)$ be the torus $Res_{E/Q}\mathbf{G}_m$. Then there's a morphism $\...

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### Sheaf of elliptic curves up to isogeny

For a scheme $X$, denote by $\mathcal{Ell}_X[\text{isog}^{-1}]$ the category of elliptic curves on $X$ localized at isogenies. Consider the functor
$$
\mathcal{Ell}^{isog}:Sch/S^{op}\rightarrow \text{...

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### Isomorphism between $\operatorname{End}_0(E)$ and $\operatorname{End}_0(E')$ as Lie algebra bundles

This may be a stupid question.
I'm reading this paper of Indranil Biswas, Tomas L. Gomez, V. Munoz (arXiv link). I have a problem in the proof of Theorem $5.3$.
Let $X$ be a smooth projective curve ...

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### rational Hodge structure of spectral curve and Prym variety

I have a problem about rational Hodge structure of spectral curves and Prym varieties.
I want to prove the following,
Let $X$ be smooth projective curve over $\mathbb{C}$, $\mathscr{M}$ be moduli ...

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174 views

### Definition of Hitchin map

This may be a dumb question.
$\mathcal{M}(r,d)$ is a coarse moduli scheme for semistable pairs $(E,\phi:E \rightarrow K_X \otimes E)$ of rank $r$, degree $d$ on a smooth projective curve $X$ over $\...

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145 views

### Hitchin map and vector bundles

I've been learning a bit about automorphisms of moduli spaces of vector bundles and the Hitchin map.
I'm reading this paper of Indranil Biswas, Tomas L. Gomez, V. Munoz, and I have a problem about ...

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### Boundary of Siegel modular variety

The moduli space of curves has a compactification whose boundary can be understood as the product of moduli spaces of curves of lower genus. Therefore (perhaps naively) one might hope that there ...

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### The Weil pairing on a generalized elliptic curve

Now I'm trying the section 6 (and 3.20) of chapter IV of Deligne-Rapoport's "Les schemas de module de courbes elliptiques".
I can't understand what $e_n$ (of 6.5.(d)) is.
It seems to be the ...

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98 views

### Berkovich Integration on algebraic curves

Berkovich developed a theory of integrating one-forms on his analytic spaces in his book "Integration of One-forms on $P$-adic analytic spaces". As this book is difficult to digest for me, I ...

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### On definition of stable vector/Higgs bundle

Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as
$\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...

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### Family over the coarse moduli space of curves

Let $k$ be an algebraically closed field. As the coarse moduli space of curves $M_g$ of genus $g$ over $k$ is not a fine moduli space, it does not have a universal family. But I am wondering if it has ...

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### A silly doubt on Log structures

Let $X=\operatorname{Spec} A$ be an affine variety. Consider the log structure given by $\mathbb N\rightarrow A$ which sends $1\mapsto 0$. Also consider the log structure $\mathbb N^r \rightarrow A$ ...

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### Non-uniruled connected smooth fibers implies flat

Let $f:X\to Y$ be a surjective morphism of connected smooth projective varieties over an algebraically closed field.
Assume all fibers are connected smooth and none are uniruled. Is $f$ flat?
In ...

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### Use of Flattening Stratification part 2 (Nitsure's construction of Hilbert and Quot schemes)

I study Nitin Nitsure's paper Construction of Hilbert and Quot Schemes (arXiv:math/0504590) and have some problems with the content of imposed universal property (F) in the section "Use of ...

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### Logarithmic Darboux theorem

Let $X$ be a smooth complex analytic manifold and $D$ be a normal crossing divisor. Suppose that there is a complex analytic logarithmic symplectic structure on $X$.
Is there a Darboux like theorem ...

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110 views

### Exterior derivative on singular analytic space

Let $X$ be a closed sub-analytic space of a smooth manifold $M$. Assume that $X$ singular. Locally at a point $x\in X$ the functions on $X$ can be extended to a neighborhood in $M$. Let $f$ be a ...

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### How to show a contraction of singular moduli space is projective?

Let $\mathcal{H}$ be a certain kind of Hilbert scheme of curves on some smooth projective variety $X$ and $\mathcal{H}$ is projective and irreducible of dimension $3$. There is a divisor $\mathcal{D}\...

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### Absolute Galois group of Q and stratification of moduli space of curves

This is slightly related, but distinct from, a question I asked earlier.
The moduli space of ribbon graphs with metric (with all vertices having degree at least 3) is isomorphic to the moduli space of ...

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### Exponential map of moduli space

Let $\mathcal{M}$ be a projective smooth moduli space over $\mathbb{C}$ (the specific example I have in mind is the moduli of curves $\mathcal{M}_g)$. Consider a point $[X]\in \mathcal{M}(\mathbb{C})$....

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### Kontsevich's A-infty cohomology classes of the moduli space of curves

In his paper "Feynman diagrams and low-dimensional topology," Kontsevich attaches to each $A_\infty$ algebra a cohomology class (with complex coefficients) on the moduli space of smooth, ...

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### Holomorphic maps on moduli space and Deformation theory

Let $\mathcal{M},\mathcal{F}$ be the classifiying spaces (i.e. complex manifolds) of two (possibly) different moduli problem. To give a map
$$f:\mathcal{M}\rightarrow \mathcal{F}$$
means that for each ...