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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30 views

Geometry of the complex Gauge group

Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$. Is there a way to endow $\...
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Pro-representability and the obstruction to deformations of “stable curve of genus one + a section”

I have 2 questions about the theorem III.1.2 of Deligne-Rapoport's "Les shemas de modules de courbes elliptiques". 1. Let $k$ be a field, $\Lambda$ a complete noetherian local ring with the ...
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Operation on algebraic stack

Let $X$ be a smooth algebraic stack and $D$ be a normal crossing divisor. Let $\pi: \tilde{D}\rightarrow D$ be the normalization of $D$. Let $\sigma: \tilde{D}\rightarrow \tilde{D}$ be an involution. ...
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166 views

Non-unique completion of a flat family of smooth projective varieties

Let $\mathbb{k}$ be an algebraically closed field of characteristic 0. Denote $S=\mathrm{Spec}\:\mathbb{k}[t]$, $U=\mathrm{Spec}\:\mathbb{k}[t, t^{-1}]$, $Z=\mathrm{Spec}\:\mathbb{k}[t]/(t)$. What is ...
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121 views

Pullback of boundary divisors under forgetful maps

Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n-$pointed smooth rational curve of genus zero and $\overline{\mathbf{U}}_{0,n}$ the universal family described by $\pi_n:\overline{\...
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Flatness of the Hitchin system?

The Hitchin fibration is a central topic of study in modern geometry. It seems to be folklore knowledge that the morphism from the coarse moduli space of semi-stable Higgs bundles to the Hitchin base ...
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$\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties

Reading M. Hindry and J. H. Silverman (Diophantine Geometry-An Introduction), I find the claim that $\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties. Mumford ...
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354 views

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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124 views

Canonical connection on $\mathcal{A}\times X$

Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
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A complex analytic version of the eigencurve

I am very much a beginner to the theory of eigencurves so there might be many mistakes in what follows, especially since it is all very speculative. My understanding of the eigencurve $\mathcal C_{N,...
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Explicit check of the invariance of the Weyl-Petersson form

Using Fenchel-Nielsen coordinates, the Weyl-Petersson metric can be written as $\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$ where $i$ is an index labelling the curves of a pants decomposition ...
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On the moduli stack of abelian varieties without polarization

(I am especially interested in abelian surfaces and characteristic 0). How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-...
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166 views

Is there a precise relationship between the goals of moduli theory and the minimal model program?

I want to get into some of the big classification problems in algebraic geometry, but have a very broad question. Ultimately we would like to classify all varieties over some field up to isomorphism, ...
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Two components on the moduli of stable maps $\overline{\mathcal{M}}_{2,0}(\mathbb{P}^1,3)$

I have two questions adapted from Exercise 24.3.1 in the book Mirror Symmetry. Consider the moduli of stable maps $\overline{\mathcal{M}}_{2,0}(\mathbb{P}^1,3)$, where a typical element is a degree $...
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Symmetric spaces as the moduli spaces of Arakelov vector bundles

Over a function field of a curve $K = k(C)$, there is the Weil uniformization $$\mathrm{Bun}_{GL_n}(C) = GL_n(K) \backslash GL_n(\mathbb{A}_K) / GL_n(\mathcal{O}_K).$$ This equality is (for example)...
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Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures

In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to ...
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Descent of projective bundles

A problem studied in GIT is the descending of vector bundles (or more in general coherent sheaves) to quotients. It is a result of Kempf that whenever we have a vector bundle over a quasiprojective ...
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Reference request: Derived structure on the moduli stack of Higgs bundles

I am reading arXiv:1708.08124. When talking about the moduli stack of Higgs bundles on a projective curve $X$. It is said on page 59, first paragraph that It is often better to put derived ...
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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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74 views

The space $M_g$ with the complex structure induced from $T_g$ is a coarse moduli space for compact Riemann surfaces of genus $g$

Proposition: The space $M_g$ with the complex structure induced from $T_g$ is a coarse moduli space for compact Riemann surfaces of genus $g$. In the proof of (1), I wonder why the holomorphic map $\...
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230 views

Derived category of singular varieties

Let $X$ be a projective variety with only normal crossing singularity. Is there a description of the derived category or the category of perfect complexes? What about the existence of semiorthogonal ...
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247 views

Relative logarithmic cotangent bundle

Let $\mathcal X \rightarrow S$ be a flat family of projective varieties over a discrete valuation ring $S$ such that the generic fibre $\mathcal X_{\eta}$ (say) is smooth projective variety and the ...
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Presentation of $H^2(\overline{M}_{0,n},\mathbb{Z})$ as an $S_n$-module?

Let $\overline{M}_{0,n}$ be the moduli space of genus zero curves with $n$ marked points. Let $I=\{\{S,S^c\}|S\subset\{1,\dots,n\},|S|\geq2, |S^c|\geq2\}$ be the set of partitions of $\{1,\dots n\}$ ...
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Getzler's stable graphs for modular operads

In The semi-classical approximation for modular operads, Getzler displays a table at the bottom of page two enumerating certain stable graphs. (This is related to the MO-Q "Stable graphs: Feynman ...
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Riemannian metric over moduli space of Riemann spheres with n punctures

In the paper `Tessellations of moduli spaces and the mosaic operad' by Devadoss (https://arxiv.org/pdf/math/9807010.pdf), on page 5-6, the author identifies hyperbolic planar tree space (or the ...
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Unirationality of universal Jacobian over special strata of moduli space of pointed genus 3 curves

Let $M_{3,1}$ be the (coarse, non-compactified) moduli space of genus $3$ curves with a marked point over a field $k$ of characteristic zero. Throwing away the hyperelliptic curves, take the open ...
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221 views

Explicitly computing Donaldson-Thomas invariants (of CY 3-folds)

I am interested in the explicit computation of generating functions of rank 1 and higher rank Donaldson-Thomas (DT) invariants. In particular, I am interested in DT invariants of K3 fibered Calabi-Yau ...
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227 views

vector bundles over projective line over an affine line

Let $k$ be a field and $E$ be a vector bundle over $\mathbb{P}_{k}^{1}\times\mathbb{A}_{k}^{1}$, does it extend to $\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}$?
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212 views

Multisections of the universal curve

Fix some $g \geq 2$. Let $\mathcal{M}_g$ be the moduli space of smooth genus $g$ curves over $\mathbb{C}$. For some $d \geq 1$, let $X_{g,d} \rightarrow \mathcal{M}_g$ be the family whose fiber over ...
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What are the genus 4 curves with Jacobians that are 4-th powers?

Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...
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Is the determinant map $det:\mathcal{M}(r,d)\rightarrow Pic^d(X)$ on moduli space of semistable vector bundles a fibration?

Let $X$ smooth projective curve over $\mathbb{C}$, fix a line bundle $L$ of degree $d$, and let $\mathcal{M}(r,d)$ denote the moduli space of semistable vector bundles of rank $r$ and degree $d$. It ...
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How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that ...
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Eichler Shimura in higher genera

Let $\mathcal{M}$ denote the moduli stack of smooth elliptic curves over $\mathbb{C}$. There is a local system, $\mathcal{H}$, on $\mathcal{M}$ with fibre $H^{1}(E,\mathbb{C})$ at the $\mathbb{C}$-...
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126 views

Rigidity of universal abelian variety

When $\phi \colon A^{\mathrm{univ}} \to \overline{A_g}$ denotes the universal Abelian variety over a compactified Siegel moduli space $\overline{A_g}$, does there exist non-trivial automorphism $\...
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1answer
157 views

The fixed points set of the actions of $\mathbb{C}^*$ and $S^1$ on the Higgs bundle moduli space

Let $\mathcal{M}_{d}(G)$ be the moduli space of $G$-Higgs bundles. $\mathcal{M}_{d}(G)$ have a non-trivial $\mathbb{C}^{*}$-holomorphic action by multiplication of the Higgs field, $$ z\cdot (E, \...
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1answer
124 views

Definition of Canonically polarized manifold?

Does anyone have a reference for the definition of a canonically polarized manifold? Typically, at least from what I have seen, a polarized manifold is a compact Kähler manifold $X$ together with an ...
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1answer
398 views

Is $H^i(\mathcal{M}_g,F)$ necessarily finite dimensional for a coherent sheaf $F$?

Let $\mathcal{M}_g$ be the moduli stack of smooth genus $g$ curves. Let $F$ be a coherent sheaf on $\mathcal{M}_g$. Is $H^i(\mathcal{M}_g,F)$ always finite dimensional? For example, $F=(f_*\Omega^1_{...
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Why not add cuspidal curves in the moduli space of stable curves?

Let $\mathcal{M}_{g,n}$ be the moduli space (stack) of stable smooth curves of genus $g$ with $n$ marked points over $\mathbb{C}. $ It's known that by adding stable nodal curves to $\mathcal{M}_{g,n}$,...
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323 views

The virtual fundamental class as derived intersection

Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
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420 views

Roadmap to understanding Gromov's Non-squeezing theorem

I'm a graduate student starting out to venture into the areas of Symplectic Geometry/Topology, and was somewhat motivated by the essence of Gromov's non-squeezing theorem which in a sense made me feel ...
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59 views

Image of Obstruction Map for Relative Quot-scheme

Let $f: X \to S$ be a projective morphisms between finite-type projective schemes, and $\mathcal{O}_{X}(1)$ an $f$-ample line bundle. Given an $S$-flat coherent $\mathcal{O}_{X}$-module $\mathcal{H}$ ...
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2answers
144 views

Nahm's equations with poles and conservation of characteristic polynomial

In "Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory" Nahm's equations are studied in Section 3. In particular, it is explained that their moduli space of solutions, $\mathcal{M}$, is ...
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1answer
237 views

On Thurston's triangulations of sphere

I have two questions from Thurston's paper [1]. In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...
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120 views

Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields

Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...
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294 views

Road map for moduli space/moduli problem/moduli stack

I am familiar with (most of the) contents of Angelo Vistoli's notes on Descent theory (Stacks). I am also comfortable with basics of Schemes, their Cohomology (Cech), from Hartshorne's Algebraic ...
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2answers
414 views

Moduli of curves over finite field

This is related to this question. I learnt about moduli problem mainly with the book Harris and Morrison. Therefore, I have only seen the construction of moduli spaces $M_{g}$ over $\mathbb{C}$. But ...
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1answer
171 views

Total Chern Class of Hodge Bundle via CohFT

I am interested in the calculation of total Chern class of Hodge bundle. I am aware that there is a way by the Grothendieck-Riemann-Roch formula, however, I read that this is also a cohomological ...
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205 views

Stacks in moduli spaces of sheaves research

I want to get some practice and build more appreciation for the use of stacks in the context of classical moduli spaces of sheaves. Here by classical I vaguely mean hands-on description of the ...
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1answer
276 views

Purity and skyscraper sheaves

In "The Geometry of moduli spaces of sheaves" a coherent sheaf $\mathcal{F}$ is defined to be pure of dimension $d$ if dim$(\mathcal{E})=d$ for all non-trivial proper subsheaves $\mathcal{E} \subset \...
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118 views

Coarse underlying curve of a smooth stable curve

In the theory of moduli spaces of smooth stable curves with $n$-marked points, I have come across the notion of the coarse underlying curve. Let $C$ be a smooth stable genus $g$ curve with $n$-marked ...

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