All Questions
Tagged with moduli-spaces ag.algebraic-geometry
717 questions
1
vote
0
answers
87
views
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 ...
- 1,982
2
votes
0
answers
125
views
On non-abelian Lefschetz hyperplane theorem
This paper studies the maps of the form $Hom(X,Y)\rightarrow Hom(D,Y)$ (where $D$ is an ample divisor on $X$) and gives conditions that when it is an isomorphism. This is called non-Abelian Lefschetz ...
- 5,901
3
votes
0
answers
213
views
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 ...
- 5,901
4
votes
1
answer
727
views
$\text{Bl}_{\Delta}(X\times X)$ 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 ...
- 1,803
1
vote
0
answers
120
views
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 ...
- 31
2
votes
0
answers
154
views
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 ...
- 1,982
2
votes
0
answers
170
views
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 ...
- 31
6
votes
2
answers
480
views
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}}...
- 1,286
2
votes
0
answers
164
views
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 $\...
- 1,982
3
votes
0
answers
125
views
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 &...
- 1,061
15
votes
3
answers
806
views
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 ...
- 3,599
4
votes
0
answers
136
views
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 ...
user168611
1
vote
0
answers
155
views
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 ...
- 195
2
votes
2
answers
443
views
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 $S_i$, ...
- 1,286
1
vote
1
answer
305
views
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 $\...
- 1,061
2
votes
0
answers
160
views
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 ...
- 31
3
votes
2
answers
578
views
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 ...
- 1,061
1
vote
0
answers
114
views
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)...
user149914
1
vote
1
answer
168
views
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 ...
- 8,998
2
votes
1
answer
383
views
$G$-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 \...
- 45
8
votes
2
answers
568
views
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 ...
- 1,061
3
votes
0
answers
166
views
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 ...
- 31
2
votes
1
answer
226
views
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=\...
- 195
4
votes
1
answer
392
views
$Ext$-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 ...
- 175
1
vote
1
answer
248
views
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 ...
- 63
3
votes
1
answer
281
views
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 ...
- 1,061
3
votes
1
answer
348
views
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,...
- 1,061
5
votes
1
answer
589
views
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 $\pi \colon \mathcal C_g \to \mathcal M_g$,...
- 445
1
vote
1
answer
291
views
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 $...
- 8,998
5
votes
1
answer
418
views
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 $Gr(1,n)$ ...
- 8,998
5
votes
1
answer
116
views
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 ...
- 53
3
votes
1
answer
252
views
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 ...
- 8,998
1
vote
0
answers
131
views
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, ...
- 123
2
votes
1
answer
149
views
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 ...
user125056
3
votes
0
answers
198
views
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 ...
- 12.1k
3
votes
0
answers
233
views
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 ...
- 196
4
votes
1
answer
372
views
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 ...
- 43
4
votes
1
answer
281
views
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
$M(r,\xi)\colon$ ...
- 297
1
vote
1
answer
148
views
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 ...
- 137
3
votes
1
answer
465
views
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 ...
- 31
2
votes
0
answers
92
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 $...
- 6,018
1
vote
0
answers
79
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\...
- 297
2
votes
0
answers
179
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 ...
user114666
1
vote
0
answers
89
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 ...
- 297
3
votes
0
answers
135
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^...
- 537
1
vote
0
answers
115
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} ...
- 297
3
votes
1
answer
440
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 ...
- 7,746
1
vote
1
answer
271
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}_{...
user168627
4
votes
0
answers
98
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 ...
- 641
1
vote
0
answers
355
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 ...
- 494