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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Derived Chow varieties
I recently encountered the "Hidden Smoothness Principle" envisioned by Deligne, Drinfeld, Beilinson, Kontsevich that singularities occurring in certain moduli spaces is the consequence of ...
4
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1
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$\text{Bl}_{\Delta}(X\times X)$ as the "Hilbert scheme" of ordered two points on $X$
$\DeclareMathOperator\Bl{Bl}\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Sym{Sym}\newcommand\Sch{\mathit{Sch}}\newcommand\Sets{\mathit{Sets}}$Let $X$ be a smooth variety. The Hilbert scheme of ...
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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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Normal bundle of a Fano threefold as Brill-Noether loci
Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient ...
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Symplectic structure on moduli space of holomorphic Abelian differentials
I've heard a "symplectic structure" referred to on the moduli space of holomorphic Abelian differentials by numerous people / sources. I do not know how to interpret this - I am looking for ...
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Why is the scheme of isomorphisms of sheaves affine over the base?
Suppose $S$ a noetherian base scheme, $X \to S$ is projective and $F, G$ are coherent $\mathcal O_X$-modules. Then by EGA (7.7.8) and (7.7.9) there exists a scheme $H = \underline{\operatorname{Hom}}...
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159
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Conics on Gushel-Mukai fourfold
Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a $\rho$-conic, then $\...
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Asymptotical equidistribution of index $p$ subgroups of $\mathbb Z^2$ on the unit tangent bundle of the modular curve
Given a prime $p$ we get $p+1$ sublattices of index $p$ in $\mathbb Z^2$ (identified with $\mathbb Z[i]\subset \mathbb C$) which correspond to some points on the moduli space of such
lattices up to ...
3
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116
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Absolutely indecomposable objects and moduli space
In the setting of moduli spaces of vector bundles/quiver representations etc I've encountered a few times situations like the one that I'll try to explain thereafter .I was wondering if there's a &...
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Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?
Belyi's theorem establishes a correspondence between smooth projective curves defined over number fields and the so called dessins d'enfants which are bipartite graphs embedded on an oriented surface ...
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Parameter spaces for conic bundles
A conic bundle over $\mathbb{P}^n$ is a morphism $\pi:X\rightarrow\mathbb{P}^n$ with fibers isomorphic to plane conics. A conic bundle $\pi:X\rightarrow\mathbb{P}^n$ is minimal if it has relative ...
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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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2
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What is the pull-back of a polarization of abelian schemes over different bases?
The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in [1].
Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, ...
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1
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Quiver varieties associated to D_4
Let $Q=(I,\Omega)$ be the $D_4$ affine quiver. We choose as dimension vector $(2,1,1,1,1)$ (where $2$ is on the central vertex). As this dimension vector is indivisible, we can choose a generic $\...
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Mirzakhani's length function integration formulas and representation varieties
Mirzakhani develops a method to integrate geodesic length functions on moduli space by considering circle bundles over moduli space given by level sets of these functions. There are natural circle ...
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2
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Moduli stack of quiver representations
Let $Q$ be a finite quiver. As far as I know there's a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT ...
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0
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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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1
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Moduli spaces of horizontal curves
Let $f:X\rightarrow Y$ be a morphism of projective varieties. We may assume that $X$ and $Y$ are smooth, and $f$ is flat of relative dimension one. Fix an ample divisor $A$ on $X$.
I would like to ask ...
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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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1
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$G$-invariant morphism and coarse moduli spaces
Let $G$ be an algebraic group acting on $X$ (a finite type scheme on $k$).
A $G$-invariant $k$-morphism $f : X \rightarrow S$ is a map such that the following commute:
$\require{AMScd}$
\begin{CD}
G \...
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2
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Explicit example de Rham moduli space of connections
Let $\Sigma$ be a Riemann surface and let $n,d$ be two relatively prime integers. We can consider different moduli spaces related to those. On one hand we have:
-$M_{Dol}$ the moduli space of stable ...
3
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1
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Ricci curvature of the Weil-Petersson metric?
Let $\omega_{\text{WP}}$ denote the Weil-Petersson metric associated to a family of Calabi-Yau manifolds. That is, let $f : X \to Y$ be a surjective holomorphic map with connected fibres such that, ...
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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 ...
3
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Why is a rational divisor class on $\overline{\mathcal{M}}_g$ determined by its values on families not mapping into a given subvariety?
This question is about Exercise 3.90 on page 143 of the book "Moduli of Curves" by Harris & Morrison.
To avoid defining stacks, the authors define a "rational divisor class on 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 \...
3
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1
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Intuition for points of the moduli of objects for a dg-category
Problem summary: I'm trying to get some intuition for what the moduli space of objects for a dg-category (as in this paper by Brav and Dyckerhoff) actually looks like/how to give an alternative ...
8
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277
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Infinitely many nonempty Seiberg-Witten moduli spaces
The classic "finiteness" statement in Seiberg-Witten (SW) theory is that, for any smooth closed connected 4-manifold, there are only finitely many spin-c structures with nontrivial SW ...
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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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Picard group of moduli of principal bundles
I am looking for the Picard group of the moduli space of principal $G$-bundles for a connected reductive complex algebraic group $G$.
Is it isomorphic to $\mathbb{Z}$? If not, what can we say when $G=\...
4
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$Ext$-algebra of stable vector bundles
Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$.
Question: What can we say about the algebra structure of ...
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1
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How can I get the scheme-theoretic support of coherent sheaf on a ruled surface with linear Hilbert bipolynomial ax+by+c?
I have pure sheaves of dimension 1 on a ruled surface, in paticular the Hirzebruch surface F$_e$=P($O \oplus O(-e)$) with linear Hilbert bipolynomial $P(x, y)=ax+by+c$.
A sheaf $E$ is pure of ...
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1
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Integral locus of Hitchin morphism
Let $\Sigma$ be a Riemann surface of genus $g$. To it, we can associated $M_{Dol}$ be the Higgs moduli space of rank $n$ and degree $d$. Fo simplicity let us take $(n,d)=1$. This quasiprojective ...
3
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1
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313
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Non Abelian Hodge theory: underlying structure holomorphic vector bundles
Let $X$ be a compact Riemann surface. We fix a complex vector bundle $E$ of rank $n$ and degree $d$ (unique up to diffeomorphism). From results coming originally (I think at least) by Simpson,...
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Tangent Space of the Hodge bundle on the moduli space of curves
Let $k$ be an algebraically closed field and $\mathcal M_g$ denote the moduli space (stack) of smooth curves of genus $g$ over $k$. Using the universal curve $\pi \colon \mathcal C_g \to \mathcal M_g$,...
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1
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271
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Tangent space to spaces of maps
Let $B = \{x_1,\dots,x_{d-2},y_1,\dots,y_k\}$ be a subscheme of $d-2+k$ distinct points of $\mathbb{P}^1$, and $g:B\rightarrow \mathbb{P}^2$ be a morphism mapping $x_1,\dots,x_{d-2}$ to a fixed point $...
5
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1
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417
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Jumping conics in Grassmannians
Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{*}S$ of the tautological bundle $S$ on $Gr(1,n)$ ...
5
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1
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Connection between braided tensor categories and local systems on moduli of stable marked genus zero curves
I'm looking for references regarding an unpublished Deligne's manuscript "Une descrption de catégorie tressée (inspiré par Drinfeld)" and the subject it touches, that is described in the ...
3
votes
1
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Moduli spaces and conic bundles
The moduli space $A_2(1,8)^{\operatorname{lev}}$ of $(1,8)$-polarized abelian surfaces with canonical level
structure has a structure of conic bundle over $\mathbb{P}^2$ with a curve of degree $4$ as ...
1
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0
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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, ...
2
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146
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Degenerations of hyperelliptic coverings
Take six distinct points $p_1,\dots,p_6\in\mathbb{P}^1$ and consider the double covering $f:C\rightarrow \mathbb{P}^1$ ramified over $p_1,\dots,p_6\in\mathbb{P}^1$. Then $C$ is a smooth curve of genus ...
3
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0
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100
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Relating different parametrizations of moduli space of Riemann surfaces
I would like to understand, as explicitly as possible, how different coordinates on the moduli space of Riemann surfaces are related:
On the one hand, there is a parametrization coming from hyperbolic ...
6
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329
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Rough paths, unparametrized path space, and Kontsevich's moduli space of stable maps
Let $X$ be a manifold. Modulo reparametrization, the path space of $X$ is a groupoid $\Pi_X$. In Kapranov's "Free Lie Algebroids and the Space of Paths", Kapranov constructs an associated ...
3
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193
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Interesting stacks with affine space as coarse moduli
I am looking for examples of Deligne-Mumford stacks whose coarse moduli space is $\mathbb{A}^n$ or at least an open subscheme of $\mathbb{A}^n$ whose complement has codimension $2$. (Thus the whole ...
3
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0
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193
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Local existence of (quasi)-universal family of sheaves
Let $p : X \to S$ be a projective morphism between two Noetherian $\mathbb{C}$-schemes of finite type with connected fibres. Let $O_X(1)$ be a very ample line bundle on $X$ relative to $S$. Given a ...
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0
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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 ${...
3
votes
1
answer
309
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Infinite automorphisms in the moduli of curves
Consider the moduli of smooth curves $M_{g,n}$ (genus $g$, $n$ marked points) and its Deligne-Mumford compactification $\overline{M}_{g,n}$ of stable nodal curves (genus $g$, $n$ marked points). This ...
4
votes
1
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266
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Torelli theorem for stable vector bundle
Let $X, X^{\prime}\colon$ smooth projective curve on $\mathbb{C}$ (genus $\geq 3$),
$M(r,d)\colon$ coarse moduli of stable vector bundles with rank $r\geq2$, and degree $d$ , and
$M(r,\xi)\colon$ ...
1
vote
1
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136
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Threefolds with Kodaira dimension 2 and non-isotrivial Iitaka map
Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.
Q1. How many such threefolds exist, and ...
3
votes
1
answer
387
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de-Rham moduli space over a compact Riemann surface
Let $X$ be a smooth projective curve over $\mathbb C$ and $M_{dR}$ denote the moduli space of stable $\Lambda$-connections of fixed rank and degree $0$. Is it known whether $M_{dR}$ is a smooth ...
2
votes
0
answers
91
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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 $...