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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Reference request: Moduli spaces of bundles over singular curves
I would like to know some reference (articles, books...) about any kind of moduli spaces of any of the following objects:
vector bundles
torsion-free sheaves
principal bundles
parabolic bundles
over ...
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Has anyone studied the Prym map for double covers with two ramification points?
If $f \colon C \to C'$ is a dominant morphism of smooth projective curves, there is a norm map $f_\ast = \mathrm{Nm} \colon JC \to JC'$ between their Jacobians, and we can consider the abelian ...
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Singularities of moduli spaces of curves
Let $\overline{M}_{g,n}$ be the moduli space of $n$-pointed genus $g$ Deligne-Mumford stable curves. This is a normal projective scheme. Then
$$codim_{\overline{M}_{g,n}}Sing(\overline{M}_{g,n})\geq ...
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Singularities of the moduli stack of Calabi-Yau threefolds
Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack?
In many cases I ...
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$Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space
There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence $f^*:\...
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Picard group of $\mathfrak{M}_g$
Let $\mathfrak{M}_g$ denote the moduli stack of Riemann surfaces of genus $g$, it is a smooth complex analytic stack, and is the analytic stack underlying $\mathsf{M}_g$, the moduli stack of complex ...
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cohomology of moduli spaces
Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...
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A question about Mirzakhani et. al.'s algebraicity theorem
While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...
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There are only finitely many varieties up to deformation
Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert ...
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Given a curve, under which condition is the set of gonal morphisms finite
Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that ...
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Kodaira-Spencer Theory and moduli of curves
I was looking at a paper of Farkas and the following confusing point came up.
Let $\mathscr{M}_g$ be the moduli stack of smooth genus $g$ curves and let $\pi: \mathscr{C} \to \mathscr{M}_g$ be the ...
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Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$
Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{...
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Degeneration of curves inside a family of surfaces
We are interested in understanding how a higher genus curve in a smooth surface of general type can degenerate into the non-normal locus of the limit of a degeneration of surfaces. More precisely:
...
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Geometric calculations using Grassmann variables
Physicists seem to get huge computational value by introducing Grassmann variables and Grassmann integration into differential geometric calculations.
See: http://en.wikipedia.org/wiki/...
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Deligne Mumford Compactification of Moduli Space Of Annuli
I am reading Abouzaid's paper "A geometric criterion for generating the Fukaya category" (https://arxiv.org/abs/1001.4593), and it is claimed there, without proof, in section C.4 in the appendix (pp....
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Reference for the Hodge Bundle
For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to $...
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Methods to compute the Kodaira dimension of moduli spaces
It is known that the moduli space $\bar{M_g}$ of genus $g$ stable curves over $\mathbb C$ is of general type for $g \geq 24$ with Kodaira dimension $3g-3=\dim \bar{M_g}$.
The idea is that one can ...
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Moduli interpretation of Fargues-Fontaine curve
The Fargues-Fontaine curve is, in his schematic version, a noetherian regular scheme $X$ of dimension 1 associated to a pair $(E,F)$, where $E$ is a local field (i.e. complete w.r.t. a discrete ...
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How do I complete the sketch of proof in 'FGA explained', 5.5.8?
It seems to me that the one part that is difficult to transfer to general coherent sheaves is given only one sentence: "Because we have such a common $m$, we get as before an injective morphism from ...
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Motivic fundamental group of the moduli space of curves?
Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...
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"A theory of generalized Donaldson-Thomas invariants" by Joyce & Song
Is anyone else working through this paper: A theory of generalized Donaldson-Thomas invariants, by Dominic Joyce, Yinan Song? I am trying to verifying example 6.2 (m=2 for simplicity) using only the ...
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What is the affinization of M_g?
This question is inspired by What is an example of a function on M_g? . Consider Mg, the moduli space of genus g curves, NOT compactified. When g is 3 or greater, this is not affine. Does anyone know ...
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Moduli 'space' of stacks?
In algebraic geometry, we are frequently interested in parametrizing geometric objects. Formally, parametrization of geometric objects having some property can be viewed as a functor $F:Sch\rightarrow ...
user74900
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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 ...
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Is a torsion free sheave of rank one on a reducible curve the pushforward of a line bundle on a normalization?
Let $X$ be a nodal curve, possibly reducible. Then can any torsion free sheaf of rank one on $X$ be expressed as $\pi_*(L)$, where $L$ is a line bundle on a partial normalization of $X$? This looks ...
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Is Teichmüller distance bigger than Weil-Petersson distance on Teichmüller space?
It is known that Teichmüller distance ($d_{Teich}$) on Teichmüller space is complete, whereas Weil-Petersson distance ($d_{WP}$) is not complete.
See for example the article
Wolpert, Scott. ...
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Why is the Hodge class of \bar{M_g} big and nef?
Let pi: \bar{Mg,1} \to \bar{M_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M_g} is the first ...
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Locus of complete curves on $\mathcal M_g$
Is the union of the complete curves on $\mathcal M_g$ Zariski dense? ($g \gg 0$)
I know it is hard to find higher-dimensional complete subvarieties of $\mathcal M_g$, but a quasiprojective variety ...
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Pullback along the Torelli map is an isomorphism
I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard ...
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Topology on the space of constructible sheaves
Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
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Moduli of extensions of modules
Given two modules $M$ and $N$ there is a nice scheme parametrizing extensions
$0 \rightarrow M \rightarrow E \rightarrow N \rightarrow 0$
namely $\operatorname{Ext}^1(N,M)$ or, leaving out the trivial ...
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Applications for knowing the singularities parametrized by the boundary of a moduli space
Given a moduli space $M$ of some smooth algebraic geometric object such as curves, surfaces, etc. Let $\overline{M}$ be a compactification of $M$. Then, $\overline{M}\setminus M$ introduces ...
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(nontrivial) isotrivial family of elliptic curves
I think it should be a standard procedure to construct such things, can anyone give a reference or give a hint? Can this be done over any base scheme?
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Progress on Bondal–Orlov derived equivalence conjecture
In their 1995 paper, Bondal and Orlov posed the following conjecture:
If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent ...
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Automorphisms of curves in positive characteristic
It is well known that over an algebraically closed field of characteristic zero a general curve (for an open subset of $M_g$) of genus $g\geq 3$ is automorphism-free.
Is this result still true over ...
user68440
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Sheaf (Gieseker) compactification of moduli space of vector bundles
I am given to understand that the moduli space $M_k^G$ of $G$ vector bundles with second Chern class $c_2=k$ over an algebraic curve/variety (for me a Riemann surface is enough/projective space for ...
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When is the sheaf of isomorphism classes of a reasonable moduli problem an algebraic space? a scheme?
Let $\mathcal{M}$ be a reasonable moduli problem (ie, at least a separated Deligne-Mumford stack, flat over some base scheme $S$).
Let $M$ be the sheafification of the presheaf:
$$(U\rightarrow S)\...
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Is there an analog of compactified moduli spaces(/stacks) for smooth manifolds?
This question has been inspired by an answer to the question Reference request: Topology on the space of smooth compact submanifolds; I've asked it in a comment to that answer but then decided to make ...
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Higgs bundles and stable vector bundle
Let $\mathcal M_X(r,0,K_X)$ be tha moduli space of semistable Higgs bundles over a smooth irreducible algebraic curve over $\mathbb C$. And let $E$ be a stable vector bundle of rank $r$ and degree $0$....
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on a Deformation long exact sequence of moduli space of stable maps
I am reading the book "mirror symmetry" by Hori,Katz,Klemm,etc. And I want to understand the following Deformation long exact sequence
\begin{align}
0 & \to Aut(Σ, p_1, . . . , p_n, f)\to Aut(Σ, ...
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Conjecture by Ekedahl on Weyl groups and Abelian varieties
A conjecture was made on p.14 in "Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties" by Torsten Ekedahl (late, excellent contributor to MO) and Gerard Van Der Geer concerning ...
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Do modular forms show up in the cohomology of moduli spaces of unmarked curves?
Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + ...
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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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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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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 ...
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Geometry of moduli problem in practice: how to check it is connected / irreducible / normal / reduced / locally complete interesection...?
Moduli spaces are very common and useful in the world of algebraic geometry. From the point view of functors, one can already check many geoemtric properties of it. I like examples, and you can assume ...
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Stacky proof of no elliptic curves over Z
It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...
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Genus=2 theta functions, Arnold's relation, and KZ connection
Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one ...
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Fundamental group of moduli space of K3's
According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...
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GAGA for moduli problems
In algebraic geometry moduli problems are described by a functor $F:\mathrm{Sch}^{\mathrm{op}}\to \mathrm{Set}$ and it is clear what a solution to a moduli problem is, namely a scheme X such that $F\...
