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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71 views
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 $...
6
votes
1answer
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 ...
1
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0answers
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\...
2
votes
0answers
117 views
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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0answers
64 views
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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votes
0answers
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 ...
0
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0answers
30 views
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 ...
3
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0answers
68 views
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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vote
0answers
72 views
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} ...
3
votes
1answer
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 ...
4
votes
1answer
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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0answers
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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0answers
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 ...
0
votes
0answers
98 views
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 $\...
2
votes
0answers
76 views
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\...
2
votes
1answer
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 ...
4
votes
0answers
114 views
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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votes
0answers
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 ...
2
votes
0answers
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 ...
4
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1answer
77 views
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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0answers
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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0answers
110 views
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 $\...
4
votes
0answers
77 views
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 ...
1
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0answers
103 views
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\...
6
votes
1answer
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 $\...
3
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0answers
141 views
Kodaira-Spencer map in logarithmic geometry
Can anyone provide a reference for the Kodaira-Spencer map in the logarithmic geometry setting?
2
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0answers
53 views
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 ...
9
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0answers
155 views
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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0answers
144 views
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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0answers
95 views
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 $\...
2
votes
1answer
239 views
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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0answers
43 views
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 ...
1
vote
0answers
69 views
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 ...
0
votes
0answers
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 $\...
5
votes
0answers
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 ...
8
votes
0answers
233 views
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 ...
3
votes
0answers
130 views
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 ...
5
votes
0answers
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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0answers
147 views
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}(...
4
votes
0answers
130 views
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 ...
2
votes
0answers
150 views
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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214 views
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 ...
5
votes
0answers
138 views
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 ...
6
votes
0answers
119 views
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 ...
3
votes
0answers
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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0answers
75 views
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}\...
2
votes
0answers
182 views
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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vote
0answers
142 views
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})$....
10
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0answers
186 views
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, ...
4
votes
0answers
81 views
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 ...
