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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Understanding dimension of gradient flow trees for product on Morse complex

I am trying to square my intuition with the facts and hope this question is not too vague. I am reading this paper which describes the $A_\infty$-category of Morse functions on a manifold $M$ of ...
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Moduli stack of doubly periodic complexes?

Let $\mathcal{A}$ be an abelian category. In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...
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52 views

What is the functor of points of the moduli scheme of stable sheaves?

Let $\Bbbk$ be an algebraic closed field of characteristic zero. Let $(\mathrm{Sch}/\Bbbk$ denote the category of locally Noetherian schemes. Let $B$ be a projective scheme over $\Bbbk$. Let $L$ be an ...
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92 views

A question about Galois group action on sheaves, descent theory and $\mu$-semistable sheaves

Let $f : Y \to X$ be a finite morphism of degree $d$ of normal projective varieties over $k$ of dimension $n$. In Lehn and Huybrechts' book "The Geometry of Moduli Space of Sheaves", there ...
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134 views

How to reduct the case $\mathcal Quot_{Y/k}(\mathcal F ,P)$ to $\mathcal Quot_{P^{n}/k}(i_{*}\mathcal F ,P)$

Let $X$ be a projective scheme over $k$.And let $i$: $X \rightarrow P^{n}$ be a closed embedding.Then for any coherent sheaf $\mathcal F$ over $X$ and any polynomial $P \in Q[Z]$.I can prove $\...
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1 answer
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Semistable pure dimension one sheaves of rank 1 and degree 0 on a singular curve

We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $...
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52 views

$\mu$-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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1 answer
123 views

The (expected) dimension of moduli space for complete intersection

When computing the dimension of moduli space for complete intersections of type $(a,b)$ in $\mathbb{P}^n$, what do we need to consider? In general we have the following part: $$|\mathcal{O}_{\mathbb{P}...
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0 votes
1 answer
209 views

Hilbert scheme of divisors in smooth projective varieties

Let $X$ be a smooth projective variety and $L$ be a line bundle with $H^0(X,L)\neq 0$. Let $D\in |L|$ and $p(t)$ be the Hilbert polynomial of $D$. Assume that any effective divisor $D'\subset X$ with ...
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Question regarding forgetting morphism of the moduli spaces of pointed rational curves

I am trying to understand the moduli space of pointed rational curves with $n$ marked points, $\overline{M_{0,n}}$. I have following doubts Let $\pi_i:\overline{M_{0,n+1}}\to\overline{M_{0,n}}$ be the ...
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4 votes
1 answer
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Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection

Suppose $C$ is a (singular) rational curve whose normalization $p: \mathbb P^1 \to C$ is a set-theoretic bijection. Can one understand how the compactified Jacobian of $C$ looks like? For example, the ...
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Moduli interpretation for integral models of PEL Shimura variety at parahoric level?

Kottwitz has built canonical integral models for a large family of PEL Shimura varieties, associated to a certain reductive group $G$ over $\mathbb Q$, when the structure level has the form $K = K_pK^...
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4 votes
1 answer
146 views

Stable cohomology of mapping class group with coefficients in $H^{\otimes n}$

Let $\text{Mod}_g$ be the mapping class group of a closed oriented genus-$g$ surface $\Sigma_g$ and let $H = H_1(\Sigma_g;\mathbb{Q})$. Fix some $r \geq 0$. It is known that the cohomology group $H^...
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2 votes
1 answer
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On the stack of semistable curves

This is a question related to Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology? Let $\mathcal C\rightarrow \mathcal M^{ss}_g$ be the universal curve over the ...
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3 votes
1 answer
135 views

$l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$

Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "G.Laumon - Fonctions zeta des variétés de Siegel" (Lemma 4.1) that ...
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198 views

Geometric meaning of cusps/component labels in Katz-Mazur book

In Katz-Mazur book "Arithmetic moduli of elliptic curves" there is a very short section (see image below) regarding the cusp-labels and component-labels. The set of cusps labels intuitively ...
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Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?

I need the reference to a detailed proof the following fact. Let $g\geq 2$ be an integer. Let $\mathcal M^{ss}_g: Sch\rightarrow Groupoids$ be the prestack which sends a scheme $T$ to the groupoid of ...
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6 votes
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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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3 votes
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On the normal crossing divisor of $\overline{\mathcal M}_g$

Let $g\geq 2$ be an integer. Let $\overline{\mathcal M}_g$ denote the DM stack of stable curves of genus $g$. It is well-known that the moduli stack is smooth and has a natural normal-crossing divisor ...
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4 votes
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224 views

Relative valuative criteria of properness for flat morphisms

Let $f: X\rightarrow S$ be a flat quasi-projective morphism, where $X$ is a smooth variety, and $S$ is a discrete valuation ring. Then we know that $f$ is proper morphism if and only if it satisfies ...
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2 votes
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132 views

Lifting a morphism along quotient of a group action

Let $X$ and $Y$ be complex projective varieties. Assume there is a finite group $G$ acting on $Y$ and we denote the quotient projective variety by $Y/G$. We have a morphism of $\mathcal{Hom}$-schemes ...
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3 votes
1 answer
124 views

2-orbifolds that I expect to be hyperbolic, but they're nonnegatively curved

I'm considering some complex 1-dimensional/real 2-dimensional orbifolds that I expect to be hyperbolic. However, some of them seem to be Euclidean or spherical. Any thoughts what's going on here? Here ...
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142 views

The genus of hyperplane sections

Let $S$ be a connected smooth projective surface over $\mathbb{C}$. Let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$, and let $d=\dim(\Sigma)$ be the dimension of $\...
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Moduli space with exceptional Mukai vector and tangent spaces at strictly semistable bundles

Assume we work (over $\mathbb{C}$) on a polarized K3 surface $(X,L)$ with a line bundle $M$ on $X$ such that $M^2=-6$ and $ML=0$ as well as $h^0(M)=h^2(M)=0$ and thus $h^1(M)=1$. Then $E=\mathcal{O}_X\...
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Existence of different Jordan-Holder filtrations

Assume $X/\mathbb{C}$ is a projective K3 surface. Let $\sigma$ be a (geometric) Bridgeland stability condition for $\rm{D}^b(X)$. My questions are : Is there any nontrivial example of $E \in \rm{D}^...
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Does there exist other known pair of Fano threefolds/fourfolds with residue categories being K3/Enriques?

Let $Y$ be a Gushel-Mukai threefold, we can either consider an ordinary Gushel-Mukai fourfold $X$ containing $Y$ as a hyperplane section, or we consider a special Gushel-Mukai fourfold $X'$ as double ...
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1 vote
1 answer
159 views

Semi-orthogonal decomposition of Verra threefold

Let $X$ be a Verre-threefold, which is by definition a $(2,2)$ hypersurface in $\mathbb{P}^2\times\mathbb{P}^2$, it is a Fano threefold. What is the semi-orthogonal decomposition of $D^b(X)$? It ...
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5 votes
1 answer
265 views

Relation between TQFT representations and factorizable sheaves

I am interested in the comparison between two different constructions which, as far as I can tell, are both supposed to produce rigorous constructions of Wess–Zumino-Witten conformal blocks. More ...
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1 vote
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Extension of a section of a line bundle on a family of curves to the central fibre

I will fix notation: $\Delta = \mathrm{Spec} R$ denotes a discrete valuation ring and $\Delta^*=\mathrm{Spec} K$ for $K=\mathrm{Frac}(R)$. Suppose we are given a curve $\pi:C\to \Delta$ and a line ...
11 votes
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329 views

Moduli stacks of algebraic surfaces—obstructions to existence?

The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic ...
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5 votes
1 answer
370 views

A question regarding isomorphism in cohomology for moduli space of stable bundles over a compact Riemann surface

Let $N(n,k)$ denote the moduli space of stable vector bundles of rank $n$ and degree $k$ over a compact Riemann surface $X$, and let $N_0(n,k)$ denote the moduli space where we fix rank $n$ and some ...
6 votes
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Affine GIT quotients and the excursion algebra in Fargues–Scholze

Some background: Let us fix a non-archimedean local field $E$ with residue characteristic $p$, and let $G$ be some connected reductive group over $E$. In [FS, §VIII.1.1] the authors define a moduli ...
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88 views

Relatively ample line bundles on trivial families of varieties of general type

Let $X$ be a smooth projective variety with $\omega_X$ ample. Let $C$ be a smooth curve. Can there be a relatively ample line bundle $\mathcal{L}$ on the trivial family $X\times C\to C$ such that, ...
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1 vote
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Computation of semistable model of curve over DVR of mixed characteristic

given a stable curve $C \in \overline{\mathcal{M}_{g,n}}(K)$ over a field $K$ and an DVR $R$ with fraction field $K$ and algebraically closed special point, is there a general technique to compute a ...
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1 vote
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Ex 1.1c Hartshorne Deformation Theory: Is this family flat?

This comes from 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 $A$ ...
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Behaviour of a certain full-flag bundle under a finite group quotient

Let $X$ be a smooth complex projective curve of genus at least 2, and let $\mathcal{M}(r,\xi)$ denote the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $\xi$. Let $\Gamma$ ...
3 votes
1 answer
263 views

When Hom scheme has projective components?

The Hom scheme of two projective varieties over some field is constructed as an open subfunctor of the Hilbert scheme of the product of the two schemes by Grothendieck. So it is a countable union of ...
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2 votes
1 answer
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Sheaves on families of genus 2 curves in Hassett's paper

Sorry for a maybe stupid long question but I'm reading the paper "Classical and minimal models of the moduli space of curves of genus two" by Brendan Hassett and I'm not able to unravel a ...
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Gluing of hybrid trajectories in Floer homology

In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
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Banach manifold structure on the moduli space of hybrid trajectories

I am reading the paper "On the Floer homology of cotangent bundles", (arXiv link) , by Abbondandolo and Schwarz and in page $35$ to define the isomorphism between the Morse complex and the ...
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1 vote
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Koszul cohomology and nodal curves

In M. Aprodu, G. Farkas - Koszul Cohomology and Applications to Moduli, arXiv:0811.3117 [math.AG], proof of Theorem(s) 4.5 (and 4.12), the authors constructed a semistable nodal curve $C^{\prime}$ of ...
1 vote
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80 views

Moduli of semistable sheaves as fiber bundle

Let $X$ be a smooth projective variety over a field of char 0, and $F$ a stable $\mathscr{O}_X$-module of rank $r$ and determinant bundle $Q$. There is a natural morphism $\operatorname{det}:M\to \...
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4 votes
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287 views

A local model of a Shimura variety and a local Shimura variety

I have a question about the book on p-adic geometry by Scholze and Weinstein. There are two ‘local theories of Shimura varieties’ written in it. The one is a local model of a Shimura variety. This is ...
1 vote
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138 views

Properness of Hom-schemes for finite group schemes

In SGA 3 XI proposition 3.12 (b) (https://webusers.imj-prg.fr/~patrick.polo/SGA3/Expo11.pdf) It is shown that: If $G$ is an affine group scheme over a base $S$ and $H$ is a finite group scheme over $S$...
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1 answer
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Is there a stable reduction for a family of vector bundles?

I. General Question Consider a one-parameter family of vector bundles $E_t$ on a smooth projective variety $X$ with fixed Chern character $v$. Suppose $E_t$ is Gieseker stable when $t\neq 0$ and $E_0$ ...
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1 vote
0 answers
70 views

semi-orthogonal decomposition of Fano fourfold associated to threefold

Let $Y$ be Gushel-Mukai threefold and $X$ a Gushel-Mukai fourfold containing $Y$ as its hyperplane section, the semi-orthogonal decomposition of $X$ and $Y$ are both known. Also, for cubic fourfold ...
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2 votes
0 answers
101 views

On non-abelian Lefschetz hyperplane theorem

This paper studies the maps of the form $Hom(X,Y)\rightarrow Hom(D,Y)$ (where $D$ is an ample divisor on $X$) and gives conditions that when it is an isomorphism. This is called non-Abelian Lefschetz ...
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3 votes
0 answers
156 views

Derived Chow varieties

I recently encountered the "Hidden Smoothness Principle" envisioned by Deligne, Drinfeld, Beilinson, Kontsevich that singularities occurring in certain moduli spaces is the consequence of ...
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4 votes
1 answer
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$\text{Bl}_{\Delta}(X\times X)$ as the "Hilbert scheme" of ordered two points on $X$

$\DeclareMathOperator\Bl{Bl}\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Sym{Sym}\newcommand\Sch{\mathit{Sch}}\newcommand\Sets{\mathit{Sets}}$Let $X$ be a smooth variety. The Hilbert scheme of ...
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1 vote
0 answers
110 views

Local description of the universal family $\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}$

I would like to get an understanding of the notion of geometric fibers of the universal family: $$\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}.$$ In fact Knudsen show ...
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