Questions tagged [moduli-space]
A Moduli space is a space in algebraic geometry whose points are geometric objects or isomorphism classes of these kinds of objects.
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Computing the virtual dimension of the moduli space $M(L, \beta)$
In Auroux's paper "MIRROR SYMMETRY AND T-DUALITY IN THE COMPLEMENT
OF AN ANTICANONICAL DIVISOR", in section §3 p.7 the author presents the moduli space $M(L, β)$ of J-holomorphic discs with ...
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Construction of Deligne-Mumford curve moduli space
The standard construction of $\newcommand{\PP}{\mathbb{P}} \DeclareMathOperator{\Hilb}{Hilb} \DeclareMathOperator{\PGL}{PGL} M_g$ is as follows (At least for stable curves and other technical ...
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Morphism of family of varieties determined on rational points
I am currently studying various Moduli problems and in order to check whether some families have non-trivial automorphisms, I have the strong intuition that the following should hold:
Let $k$ be an ...
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representing compactified stack as a global quotient
Let $M$ be a DM stack over a field $k$. Assume I managed to represent it as $[X/G]$ for some scheme $X$ and a group scheme $G$. Does that imply that the DM compactification $\overline{M}$ of $M$ is ...
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Modular forms defined on the modular curve
I am studying the book "Harmonic maass forms and mock modular forms theory and applications"
In section 7.5, they deal with p-adic harmonic maass functions.
At first, they define the space $...
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Neron polygons and cusps on the modular curve
I am interested in understanding the computation of the number of cusps of $X_1(N)$, which is the modular curve that parameterizes the space of pairs of a generalized elliptic curves $E$ and a point $...
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Rational points on the the reduction mod $p$ of $X_0(N)$
Consider the modular curve $X_0(N)$ over $\mathbb Q$ and for $p\mid N$ consider the reduction modulo $p$ of $X_0(N)$. Let's denote this curve with the symbol $X_p$ (we know that it is a singular curve ...
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$\mu$-polystable locally free sheaf
In Huybrechts and Lehn's book "The Geometry of Moduli Space of Sheaves",a sheaf $E \in Ob(Coh_{d,d-1}(X))$ is polystable if $E \cong \bigoplus E_{i}$ in $Coh_{d,d-1}(X)$,where the
sheaves $...
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Isomorphism in $Coh_{d}(X)/Coh_{d^{'}-1}(X)$
I'm reading "The Geometry of Moduli Space of Sheaves" (Huybrechts,Lehn).On page 26,$Coh_{d}(X)$ is defined as the full subcategory of $Coh(X)$ whose objects are sheaves of dimensions
$\le d$....
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Hat Knot Floer Homology with Z coefficients calculation
I would like to ask if there is a reference which carries out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{HFK}(K;\mathbb{Z})$, where the ...
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Hilbert scheme of $\mathbb{P}^2$ is not a product?
Fixing a connected component in the Hilbert scheme of $\mathbb{P}^2$ is the same as fixing the Hilbert polynomial $ax+b$. Taking a primary decomposition of any subscheme in this connected component ...
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An examle of a stable sheaf which is not geometrically stable
I am trying to understand an example of stable sheaf which is not geometrically stable on p.19 of Huybrechts and Lehn's book Geometry of Moduli Space of Sheaves
Let $X=Proj(R[x_{0},x_{1},x_{2}]/(x_{0}^...
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Degree of Hodge bundle
Let $H$ be the Hilbert scheme parametrizes subschemes of $\mathbb P^{5g-6}$ with Hilbert polynomial $p(m)=(6g-6)m+(1-g)$ (for example, curves with genus $g$ embedding by the canonical bundle to the ...
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What does "intersect properly" mean?
A corollary in The Geometry of Moduli Space of Sheaves (Huybrechts, Lehn) says:
Let $X$ be a normal closed subscheme in $P^{N}$ and $k$ an infinite field. Then there is a dense open subset $U$ of ...
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Why is the definition of the dual sheaf is independent of the ambient space?
I'm reading "The Geometry of Moduli Space of Sheaves" (Huybrechts,Lehn). He gives a new definition of the dual sheaf:
$\ $Let E be a coherent sheaf of dimension $d$ ,and let $c=n-d$ be its ...
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Learning about Moduli spaces of sheaves
I am a Ph.D. student and starting a side project with a fellow student on Moduli spaces. Our plan was to start with the book on Invariants and Moduli by Mukai (starting from chapter 5) and use the ...
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Relationship between GIT and coarse moduli spaces
I'm trying to understand how a generic algebraic geometer constructs coarse moduli spaces. I'm familiar with the definition, and how it is usually quite involved to show that a space has the coarse ...
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Are points $(a:b)\in \mathbb{P}^1$ which satisfy $X\subset V(af+bg)$ closed?
Let $X\subset \mathbb{P}^m_k$ be a closed irreducible subvariety, and $f,g\in k[x_0,\dots,x_m]$ be homogenous polynomials of $\deg f=\deg g$.
Then we have parametrized hypersurfaces $V(af+bg)\subset \...
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Differences between equivalent definitions of algebraic spaces
I have trouble understanding the differences between the following two equivalent definitions of algebraic spaces. Let $k$ be a field.
The first one is
An algebraic space is an '{e}tale sheaf $(Sch/S)...
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What does the lattice Λ= Z.z+Z.1 mean?
I'm currently reading The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid and I got stuck (on the fourth page) on the following paragraph:
... the quotient space $\Gamma_1$ \ $\...
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Coarse moduli space for semi-stable vector bundles on a curve.
Given a curve $C$ of genus $g$ I know how to construct a quasi-projective variety $M_C(r,d)$ as the GIT quotient of a certain Quot scheme $Q=Quot^{r,d}(\mathcal{O}_C(-n)^N)$ by the action of a certain ...
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"Universal family" used despite moduli space not being fine?
This is something that has nagged me for a while, and I think I basically know the issue, but thought I would see what people had to say.
In the study of moduli of curves, one will sometimes see the ...
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Hilbert scheme of points on surface is a crepant resolution of symmetric product
If $S$ is a smooth projective surface, it is well-known (I believe first proven by Fogarty?) that the Hilbert scheme of points $\text{Hilb}^{n}(S)$ is a smooth irreducible variety of dimension $2n$. ...
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What does $\mathbb{C}[f]$ mean?
Let $f\colon E \to E$ be an endomorphism of stable bundles over a variety $X$ over $\mathbb{C}$.
What does the notation $\mathbb{C}[f]$ mean? (This is supposed to be a field if $f$ is an isomorphism.)
...
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Why use orientation-preserving diffeomorphism (instead of all diffeo's) in the construction of the moduli space of a Riemannian manifold
The question is basically in the title, but I want to make it more precise:
Given an oriented Riemannian 2-manifold $\Sigma$ one can take a quotient of the set
$$
\mathcal{M}_+(\Sigma)=\{~c~\mid (\...
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What is an instanton? (On a complex surface or a differentiable 4-manifold )
The question is as in the title. I have browsed online (Wikipedia, etc) and while they do give me the definition, it gets a bit too much physics-y for me. Therefore I would appreciate it if someone ...
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The stack classifying non-oriented triangles is equivalent to a quotient stack $[\widetilde{T}/S_3]$
I come across problems when I try to compute the stack classifying non-oriented triangles. My reference is the book Algebriac Stacks ( K. Behrend, B. Conrad, D. Edidin, B. Fantechi, W. Fulton, L. ...
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Singularities of arithmetic surfaces
I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves.
The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
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Uniqueness of model of curve over DVR
I understood that properness of $\overline{\mathcal{M}_{g,n}}$ tells us that for a fixed field $K$, a curve $C$ over $K$ and a DVR $V$ with fraction field $K$, there is a unique curve over $V$ with ...
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Exercise 1.1c in Hartshorne's Deformation Theory: Is this family of conics flat?
In my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ($k$ is ...
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calculate the dimension of moduli space
I'm learning gauge theoretic topics about 4-manifolds, and I get stuck when I try to calculate the dimension of ASD moduli space.
For an oriented closed 4-manifold $M$, we first fix a riemannian ...
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local model of gauge theoretic moduli space
I'm a beginner of gauge theory and I find that most materials state the theorem(without proof) that:
Given a principal bundle $P$, for a connection $A$ over $P$, a isotopy group $G_A$ of $A$ consist ...
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Compactification of $M_{1, 1}$
Let $M_{1, 1, \mathbb{Z}}$ be a moduli stack of elliptic curves. Denote $M_{1, 1, k}$ its base change to a field $k$. We have the map $j: M_{1, 1, k} \to \mathbb{A}^1_{k}$ which realizes the affine ...
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Moduli space of connection on line bundle
I want to show that for a $U(1)$ bundle $P$ over a connected smooth 4-manifold $X$, the moduli space of Yang-Mills connection over $P$ is the torus $H^1(X,\mathbb{R})/H^1(X,\mathbb{Z})$.
Now I reduce ...
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Assumptions on a scheme X needed to construct the Picard scheme $\operatorname{Pic}(X)$
Let $X$ be a scheme. I believe that Grothendieck was the first to put a scheme structure on the Picard group $\operatorname{Pic}(X)$, where he assumed that X is projective and reduced. Then Mumford ...
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What do we know about the space of homomorphisms of semistable vector bundles?
Let $X$ be an algebraic curve over $\mathbb C$ (feel free to add smoothness or other usefule properties). Fix degrees $d_1,d_2$ and ranks $r_1,r_2$. The slopes are $\mu_i=d_i/r_i$. Assume $\mu_1>\...
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Étale local universal sheaves
Moduli functors for (flat families of) semistable coherent sheaves over a projective scheme are not representable functors, which is to say that no global universal family exists. However one can ...
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Reference request: Katz modular forms modulo $p$ at cusps vs classical modular forms modulo $p$ at cusps
I, like the author of this post, am severely lacking the background to make the connection between
reducing modular forms' $q$-expansions modulo $p$ at various cusps, and
$q$-expansions of modular ...
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Is a coarse moduli space whose objects have no automorphisms also fine?
We work over the complex numbers $ k = \mathbb{C} $.
Let $ F $ be a moduli problem, i.e. a contravariant functor $ F : \text{Sch}/k \rightarrow \text{Sets} $. Suppose that $ M $ is a coarse moduli ...
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Involution of Legendre Family
I'm trying to understand why automorphisms of elliptic curves lead to non-existance of the moduli space (as a variety). Some notes suggest to consider the following example:
Consider the Legendre ...
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What is the moduli space $M _{0,5}$?
Let $M_{0,n}$ denote the moduli space which consists of genus 0 non singular projective curves with n distinct marked points upto marked point isomorphism.
So $M_{0,3}$ is a singleton set as any 3 ...
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How to construct the schemes $\operatorname{Hom}_S(X, Y)$ and $\operatorname{Isom}_S(X, Y)$ from the Hilbert scheme?
Let $S$ be a noetherian scheme. Grothendieck [1] constructed for any projective $S$-scheme $X$ the Hilbert scheme $\operatorname{Hilb}_{X/S}$, which represents the functor
$$T \mapsto \underline{\...
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Moduli space of special Lagrangians
I'm currently reading Auroux's Mirror Symmetry and T-duality in the Complement of the Anticanonical Divisor and Special Lagrangian Fibrations, Wall-crossing, and Mirror Symmetry back and forth.
I'm ...
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Teichmuller space of the 4-punctured sphere
I'm a bit confused working with Teichmuller space at the moment. Let's think of Teichmuller space as the space of holomorphic/conformal structures on a surface, up to diffeomorphisms isotopic to the ...
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Notation in Katz-Mazur Arithmetic Moduli of Elliptic Curves
A moduli problem $\mathcal{P}$ is a contravariant functor $\mathbf{Ell}\to\mathbf{Set}$. The objects of $\mathbf{Ell}$ are arrows $E\to S$ from an elliptic curve $E$ to a varying base scheme $S$. The ...
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Is the moduli stack of stable maps functorial?
Let $f:X\rightarrow Y$ be a morphism where $X$ and $Y$ are smooth projective varieties over $\mathbb{C}$. Does $f$ induce a well-defined "pushforward" morphism $\overline{\mathcal{M}}_{g,n}(...
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Configuration spaces to moduli spaces
In Segal's paper on Mapping Configuration spaces to moduli spaces, I'm not understanding what the map $\Phi$ is, explicitly.
Also in section 2, he goes on to say $M_{g,2} \simeq BHomeo^{+}(F_{g,2}; \...
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moduli space of elliptic curves, $\mathbb{H}/SL(2,\mathbb{Z})$ or $\mathbb{H}/PSL(2,\mathbb{Z})$
I see some (most) people saying the moduli space of elliptic curves is $\mathcal{M}_{1,1}=\mathbb{H}/SL(2,\mathbb{Z})$ which is an ineffective orbifold. But it is also fine to forget the trivial ...
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When is $\operatorname{Spec} A^G \cong (\operatorname{Spec} A)/G$ true?
Let $G$ be a group acting on a ring $A$. I would like to know in which generality we know that $\operatorname{Spec} A^G \cong (\operatorname{Spec} A)/G$. Moreover, when this is true, it also holds for ...
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What does it mean for points of the modular curve $X(N)$ to be "defined over $\mathbb{F}_p$"?
I'm trying to study a collection of elliptic curves over some fixed finite field $\mathbb{F}_p$. By browsing the literature and discussing with my supervisor, it seems like it will be fruitful to ...
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