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.
920 questions
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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 ...
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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 $\...
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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
$$\Phi : \overline{\mathcal{X}}_{g,n} \xrightarrow{\sim} \overline{\mathcal{M}...
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$\mathbb{C}^*$-action on moduli space of Higgs bundles
Let $M_{r,d}$ be the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ over a compact Riemann surface. Over $M_{r,d}$ we have a $\mathbb{C}^*$-action $$t \cdot (E,\phi)=(E, t \phi). $...
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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 ...
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Two elliptic curves with the same j-invariants
This is an interesting observation of mine when exploring moduli of elliptic curves.
Let's fix a base field $k$ and assume that we have two elliptic curves $E$, $E'$ which are isomorphic over $\bar k$...
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Is $\mathcal H/\Gamma$ defined over a number field when $\Gamma$ is not congruent?
Let $\mathcal H$ denote the upper half plane of complex numbers, and $\Gamma$ an arithmetic subgroup of $\operatorname{SL}_2(\mathbb Z )$ (not necessarily congruent). I wonder whether $\mathcal H/\...
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Hat knot Floer Homology with Z coefficients calculation
I would like to ask for recommended references which carry out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{\operatorname{HFK}}(K;\mathbb{Z})...
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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 $C$...
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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 ...
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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 ...
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Moduli space of stable bundles with fixed Chern class is bounded
This is an excerpt on a set of lecture notes I'm going over:
Classically, a lot of interest in stable vector bundles is due to the fact that stability allows the study of moduli of vector bundles via ...
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Moduli space of complex and anti-complex tori?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}$By Will Sawin's answer to Moduli Spaces of Higher Dimensional Complex Tori the moduli space of complex $d$-tori is $X ...
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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 ...
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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 $...
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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 ...
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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 ...
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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....
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References on the claims of moduli spaces of additive compactifications by Hassett-Tschinkel
I am considering the classification problem of Artinian local $\mathbb{C}$-algebras, and notice the paper
Hassett, Brendan; Tschinkel, Yuri, Geometry of equivariant compactifications of (\mathbb{G}^...
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The smoothness of solutions to the Hitchin self-dual equations within a stable orbit after Sobolev completion
First, let me introduce the background of the problem: While studying chapter 4 of the Hitchin's paper "THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE", I encountered an issue. Hitchin ...
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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 ...
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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 ...
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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 ...
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How are moduli spaces related to geometric complexity theory?
I am interested in understanding the relationship between moduli spaces and geometric complexity theory (GCT).
Relation between moduli spaces and GCT:
How are moduli spaces related to geometric ...
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GIT semi-stability on graded Artinian local $\Bbbk$-algebras
Let $\Bbbk$ be a algebraically closed field of characteristic zero. A graded Artinian local $\Bbbk$-algebra is $(A,\mathfrak{m},\bigoplus A_i)$ such that $(A,\mathfrak{m})$ is an Artianian local $\...
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"Saddle connection" on a translation surface
A saddle connection on a translation surface $\omega$ is a geodesic
in the flat metric determined by $\omega$ joining two zeros with no zeros in its interior.
Athreya, Jayadev S., and Howard Masur. ...
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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 \...
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How to define a family of Hilbert $A-B$-bimodules $ \pi \ : \ M \to X $, parametrized by a $C^*$-algebra $X$?
Let $A$ and $B$ two $ C^* $ - algebras.
I would like to define a functor $ X \to \mathrm{Bimod}_{A,B} (X) $ which associate to any object $X$, the set of isomorphism classes of a family of Hilbert $A-...
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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 ...
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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 ...
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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 ...
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Stack of smooth fiber bundles with fiber $F$
I'd like to premise that while I know the definition of (differentiable) stack, I'm not really into the language of schemes so my understanding of what is a moduli stack is pretty concrete and ...
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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
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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 ...
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Moduli space of curves away from singular subsets
Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities.
More specifically, I'm interested in ...
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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 ...
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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 ...
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Teichmüller theory for open surfaces?
I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces?
My motivation basically is that I would like to find out more about the "...
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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 ...
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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-...
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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]} \...
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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 ...
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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 ...
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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 ...
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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(...
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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}: \...
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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 ...
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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 ...
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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{\...
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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 ...
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