All Questions
Tagged with moduli-spaces ag.algebraic-geometry
717 questions
3
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1
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197
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Square root of relative Kähler differentials and families of curves
Let $f: X \to S$ be a smooth morphism of schemes of relative dimension $1$. Then $K=\Omega^1_{X/S}$ is a line bundle on $X$. I am interested in the following question:
When does $\Omega_{X/S}$ have a ...
- 6,622
3
votes
0
answers
104
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Jacobian of a reducible curve with arbitrary singularities
Let X be a reduced, reducible curve over $\mathbb{C}$ with locally planar singularities, and let $\widetilde{X}$ be its normalization. I am interested in the Jacobian varieties $\mathrm{Jac}(X)$ and $\...
- 31
2
votes
0
answers
134
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Universal semistable curve
For the moduli of stable curve, we have the result by Deligne-Mumford-Knudsen that there's an isomorphism of moduli spaces
$$\Phi : \overline{\mathcal{X}}_{g,n} \xrightarrow{\sim} \overline{\mathcal{M}...
- 834
2
votes
1
answer
154
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$\mathbb{C}^*$-action on moduli space of Higgs bundles
Let $M_{r,d}$ be the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ over a compact Riemann surface. Over $M_{r,d}$ we have a $\mathbb{C}^*$-action $$t \cdot (E,\phi)=(E, t \phi). $...
- 1,061
3
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0
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73
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What exactly does it mean for the moduli space of stable sheaves to have a universal family étale locally?
It is known that in general, the moduli space $M$ of (Gieseker) stable sheaves do not have a universal family. Therefore, a map $f \colon S \to M$ does not necessarily induce a family of sheaves.
On ...
1
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0
answers
116
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Universal picard variety of degree d
Let $M_g$ denote the moduli space of smooth genus $g$ curves over $\mathbb{C}$, where $g \geq 2$. Let $Pic^{d,g}$ denote the universal picard variety over $M_g$, parameterizing pairs $(C,L)$ where $C$...
- 129
1
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0
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102
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weak (?) valuative criterion for properness
In the article "On the Kodaira Dimension of the Moduli Space of Curves" by J. Harris and D. Mumford, to prove that
$\overline{H}_{k,b}$ is proper over Spec $\mathbb{C}$, the authors refer to ...
- 560
4
votes
1
answer
165
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Describing the compactified Jacobian of a nodal curve
$\DeclareMathOperator{\Pic}{Pic}$Let $C$ be an integral projective curve over $\mathbb C$, which is smooth except for a single node $x\in C$. Let $M$ be the moduli space of stable torsion-free sheaves ...
- 1,286
3
votes
0
answers
197
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Concrete reasons to study derived categories of quasi-coherent sheaves on algebraic stacks?
Title says it all. There seems to be a lot of work done on derived categories of quasi-coherent sheaves on stacks, and yet I couldn't find many "external" applications. I'm mainly wondering ...
- 143
0
votes
0
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98
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Differential of the evaluation map of the Kontsevich moduli space
Let $X$ be a smooth projective variety, and $\beta$ a curve class on it. We have the Kontsevich moduli space $\overline{\mathcal M}_g(X, \beta)$ of stable maps from genus $g$ curves to $X$ with class $...
- 19
2
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0
answers
103
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Representability of stack of finite maps between curves
I am interested in the following moduli problem: The moduli functor $\mathcal{F}$ has $T$-points:
a nodal $n$-pointed curve $C/T$ of genus $g$.
a nodal $b$-pointed curve $D/T$ of genus $h$.
a finite ...
- 223
5
votes
0
answers
175
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Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?
In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...
- 641
1
vote
1
answer
127
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Vanishing of higher morphisms for pair moduli
Consider a moduli scheme $P$ of sheaves $F$ on a Calabi-Yau n-fold $X$ with a section $s\in H^0(F)$. Alternatively, such objects can be described as maps $\mathcal{O}_X \xrightarrow{s} F$ called pairs....
- 988
2
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0
answers
171
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A conjecture on the scheme-theoretic image of a moduli map
Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k$, and let $K'/K$ be a finite tamely ramified Galois extension with residue field $k'$. Let $E/\mathbb{Q}_p$ be a sufficiently large ...
2
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0
answers
101
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Conjecture on the moduli space of stable sheaves on Calabi-Yau threefolds
Let $X$ be a smooth projective Calabi-Yau threefold over $\mathbb{C}$. Let $M_{X}(r,c_1,c_2)$ denote the moduli space of Gieseker-stable sheaves on $X$ with Mukai vector $(r,c_1,c_2)$.
Is the ...
1
vote
0
answers
60
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Distribution of the marked points on the components of a stable n-pointed curve of genus zero
Let $\overline{M}_{0,n}$ be the fine moduli space of stable n-pointed curves of genus $g=0$. Let $[(D_{0},p_1,...,p_n)] \in \overline{M}_{0,n}$. Suppose that each component of $D_0$ contains at least ...
- 560
2
votes
1
answer
217
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Dual of slope semistable vector bundle on higher dimensional variety
Recall the definition of slope semistability, taken from section 1.2 of Huybrechts and Lehn's "Geometry of Moduli Spaces of Sheaves" book. Let $X$ be a projective $\mathbb{C}$-scheme and $E \...
- 129
8
votes
0
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333
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Do automorphisms actually prevent the formation of fine moduli spaces?
I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
- 821
1
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0
answers
115
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Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12.1k
1
vote
0
answers
66
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Non-empty chambers for Hassett spaces
Fix some $n\geq 4$. Hassett constructed different compactifications of $M_{0,n}$ that depend on the input data of what he calls collections of weight data, which are elements of the set of admissible ...
- 650
1
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0
answers
133
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Question about wall crossing for Hassett spaces
The context of this question is that one of Hassett's famous compactifications of $M_{0,n}$ by means of weighted stable marked curves. I imagine the answer to my question is well known, but I haven't ...
- 650
2
votes
0
answers
139
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universal structures over $\mathcal A_g$
Over the moduli space of curves, $\overline{\mathcal{M}}_{g,n}$ there are several "natural" spaces like the universal curve, the universal Jacobian, the space of stable maps, the universal ...
2
votes
1
answer
197
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Books and lecture notes about Moduli spaces of Abelian varieties
Following this question, I would like to ask about books and lecture notes for Moduli spaces of Abelian varieties. I suppose that Mumfords book "Geometric Invariant theory" treats it but it ...
- 224
2
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0
answers
233
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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 ...
- 91
2
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0
answers
70
views
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 ...
- 223
2
votes
1
answer
207
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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-...
- 41
1
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0
answers
84
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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]} \...
- 209
1
vote
0
answers
94
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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 ...
- 161
2
votes
0
answers
127
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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 ...
- 1,726
3
votes
0
answers
152
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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 ...
- 984
3
votes
1
answer
250
views
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(...
- 3,625
2
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0
answers
98
views
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}: \...
- 223
1
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0
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250
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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 ...
- 161
4
votes
0
answers
515
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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 ...
- 161
2
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0
answers
155
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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{\...
- 223
1
vote
0
answers
180
views
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 ...
- 426
2
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0
answers
102
views
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 ...
- 285
4
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0
answers
152
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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 ...
- 1,790
1
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0
answers
137
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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 $...
- 984
1
vote
0
answers
145
views
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 ...
- 395
1
vote
0
answers
138
views
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 ...
- 930
6
votes
4
answers
673
views
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 ...
- 153
1
vote
1
answer
184
views
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 ...
- 123
2
votes
1
answer
293
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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{...
- 5,701
1
vote
0
answers
47
views
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 ...
- 195
2
votes
1
answer
182
views
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 $\mathcal{O}_X$. ...
- 930
7
votes
0
answers
270
views
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)) = \...
- 131
1
vote
1
answer
254
views
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:/...
- 163
2
votes
0
answers
193
views
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 ...
- 61
2
votes
0
answers
113
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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}^{...
- 223