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 ...

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### The Deligne-Mumford Compactification for Closed Surfaces

I am reading this note on super-Riemann surfaces. In the second paragraph of section 7.4.1 (page 87), there is a statement that I am trying to understand:
The compactified moduli space of closed ...

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### Fine moduli space of rigid stable families to $P^r$

In FP-notes. in Theorem 3, pages 16-17, in order to prove that a functor of rigid stable families (in genus zero) has fine moduli space, they defined a $H$-balanced morphism and they show that we have ...

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### Flatness properties of Poincare sheaf on relative compactified Jacobians

Assume we are given a $K3$-surface $X$ with $Pic(X)=\mathbb{Z}\mathcal{O}_X(1)$ such that $\mathcal{O}_X(1)$ ample and of degree 2.
Then we have the linear system $|\mathcal{O}_X(1)|=\mathbb{P}^2$ ...

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### Semistable vector bundles over the complex projective line are trivial bundles

I currently found some argument about semi-stable vector bundle over the complex projective line, which says that
"Over the complex projective line, a (holomorphic) vector bundle is semi-stable if ...

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378 views

### Prerequisites for reading papers of arithmetic such as Ribet, Mazur, Faltings, Wiles

I've studied some fundamentals of algebraic geometry and number theory, and now I want to read papers which seem to be the "main stream" of frontier research on arithmetic.
I've heard that Mazur's "...

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### Is a Procesi bundle equipped with a hyperholomorphic connection?

Haiman has constructed in the paper the unusual tautological bundle $P$, called Procesi bundle, of rank $n!$ over the Hilbert schemes of points on the affine plane in the following way.
Let $H _ { n }...

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### Moduli space is a Calabi-Yau manifold?

I asked a question here where a moduli space of flat connection is related to the $n$-dimensional complex projective space:
$$\Bbb E/S_n \cong \Bbb P^{n-1}. $$
This is related to a 4d SU(N) Yang-...

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### The construction of moduli space of stable maps

In FP-notes for proving existence of $\overline{M}_{0,n}(P^r,d)$ we first defined $\bar{t}$-rigid stable family and found fine modulo space for it where $\bar{t}=(t_0,...,t_r)$ is basis for $H^0(P^r,O(...

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### The Precise Meaning of the Moduli Space of Flat Connections?

Questions: I would like to have a precise description of the meanings of the Moduli Space of Flat Connections, such that it is understandable by mathematical physicists and physicists.
For 3d Chern-...

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### Derived functor giving algebraic map between moduli

Let $X$ be a smooth variety and $M$ a fine moduli space of certain kind of sheaves on $X$. Let $\mathcal{E}$ be the universal family on $X\times M$. Suppose there is a derived functor $F$ from $D^b(X)$...

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### How to construct $ \mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \} $?

Set : $$ \mathcal{H} ( \mathbb{P}^n ) = \{ \ \text{Smooth projective subvarieties of } \mathbb{P}^n \ \} $$
I would like to know if there exists a projective variety $ H ( \mathbb{P}^n ) $ whose ...

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### Maps between moduli of curves

Let $M_{g,n}$ be the moduli space of $n$-pointed curves, and $M_g[m]$ the moduli space of (unpointed) curves with $m$-level structure.
Fix $m>0$. Is it true that for $n$ large enough, there is a ...

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### Is the stack of stable curves with no rational component algebraic?

Let $g\geq 2$ be an integer and let $\overline{\mathcal{M}}_g$ be the (smooth proper Deligne-Mumford) algebraic stack of stable curves of genus $g$.
Let $\mathcal{M}_g^{nr}$ be the substack of ...

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### Moduli space of linear partial differential equations

Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities?
This is in connection with a quote from someone on the web that I saw a long time ago. At ...

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### Good introductory references on moduli (stacks), for arithmetic objects

I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and ...

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### How to calculate : $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) $?

I try to calculate the rational cohomology algebra $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) = \displaystyle \bigoplus_{k=0}^{+ \infty} \mathrm{Hdg}^{ 2 k } (...

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### The ¨irreducible¨ representation variety of surface group

Let S be a closed surface of genus larger than 1, G be a compact, simply connected simple Lie group with finite center. Consider the representation variety M(S,G)=Rep($\pi_1$(S), G). Witten´s Formula ...

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### Confusion in known result about Moduli space of vector bundle of rank 2 degree 0 vector bundles over smooth curve of genus 2

Theorem: Let $X$ be a complete, non-singular algebraic curve of genus
$2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector
bundles of rank $2$ and degree $\Theta$. The ...

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330 views

### What are some open problems in moduli spaces and moduli stacks?

I would like to know what are the open big and interesting problems related to moduli spaces and moduli stacks ?
Thanks in advance for your help.

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### Computing $H^*(BDiff(W_{\infty},D^{\infty});\mathbb{Q})$ via Mumford-Morita-Miller classes

Galatius and Randal-Williams proved the following generalized Mumford conjecture in their joint paper, "Stable Moduli spaces of High Dimensional Manifolds". For each characteristic class of oriented $...

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### Generalizing the Madsen-Weiss Theorem via the scanning map $\mathscr{C}(M,\mathbb{R}^{\infty})\to\Omega^{\infty}AG^+_{\infty,d}$

The Madsen-Weiss Theorem, as described by Hatcher, states that there is an isomorphism $H_*( \mathscr{C}_{\infty})\cong H_*(\Omega_0^{\infty}AG^+_{\infty,2})$ where $\Omega_0^{\infty}AG^+_{\infty,2}$ ...

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### Virtual fundamental class of Moduli space of stable maps in genus 1

What is the virtual fundamental class of $\overline{M}_{1,n}(\mathbb{P}^2,d)$? In general the virtual fundamental class is difficult to compute I guess. But if you look at Proposition 2.5 of https://...

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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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### 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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### Stable torsion free sheaf on smooth projective surface

Let $E$ be a torsion-free sheaf on a smooth projective variety $X$ over $\mathbb{C}$. Let $H$ be an ample line bundle on $X$. Then we say $E$ is stable if $\mu_{H}(F)<\mu_{H}(E),\,\forall 0\neq F \...

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### When is the moduli of generalized parabolic bundles with fixed determinant smooth?

Let $X$ be a smooth, projective curve of genus at least $2$, $x_1, x_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic ...

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### Rationality of the moduli space of genus g curves

I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions ...

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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 ...

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### Small resolution and Intersection homology

Let $X$ be a variety and $Y\rightarrow X$ be a small resolution. Then an old result by Goresky Macpherson says that the intersection homology of $X$ is same as the usual betti homology of $Y$.
My ...

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### Moduli of stable and semistable $G$-Higgs bundles on curves

Let $X$ be an irreducible smooth projective curve of genus $g \geq 2$ over $\mathbb{C}$. Let $G$ be a connected reductive affine algebraic group over $\mathbb{C}$. Let $\mathcal{M}_{G,Higgs}^s$ (resp.,...

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### Tangent and cotangent bundle of a smooth algebraic stack

Are there any good notion of tangent and cotangent bundle (and stacks) of a smooth algebraic stack, similar to the notion of tangent and cotangent bundles (and spaces) of smooth schemes?
I am ...

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170 views

### How to split a Multi-section into finitely many Sections via base-change?

Let $:f:X\to Y$ be a projective surjective morphism between two normal varieties over $\mathbb{C}$. Assume that $f$ has only $1$-dimensional fibers. Let $D$ be a multi-section of $f$, i.e., $D$ is a ...

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138 views

### Metric on moduli space of semistable principal G-bundles on curves

Let $X$ be an irreducible smooth projective curve over $\mathbb{C}$. Let $G$ be a connected reductive linear algebraic group over $\mathbb{C}$. Let ${\rm M}_{G,X}$ be the moduli space of semistable ...

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### Moduli problem of stable nodal curves over the integers

Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}$ represents the functor $$\overline{\mathcal{M}}_{g,n}(...

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### Generic Mumford Tate group and algebraic points

I will stick with a concrete example for this question, but it should probably be cast in a more general framework.
Let $Sym_g(\mathbf{C})$ be the space of symmetric matrices of order $g$ with ...

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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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### Galois representations associated to the modular tower and automorphy

Consider $\mathcal{M} = \{M_{g,n}, \mu_{g,n}^{g’,n’}\}$, the system of moduli spaces of $n$-pointed smooth algebraic curves along with the basic maps amongst them coming from identifying and ...

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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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### On a presentation of $\bar{M}_{2,1}$

In this answer it is said that $\bar{M}_{2,1}\cong \bar{M}_{0,7}/S_6$. However, I cannot see this. Given a curve of genus $2$ and a marked point, quotienting by the involution surely gives a rational ...

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### A log structure on the moduli space of curves

Let $M_{g, n}$ be the moduli space of curves of genus $g$ with $n$ marked points. Let $M_{g, \vec{n}}$ be the moduli space of marked curves with a choice of a (possibly zero) tangent vector at each ...

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### determinant of a coherent sheaf, locally free on a big open set

Let $f:X\rightarrow S$ be a smooth projective morphisms of noetherian schemes and let $\mathcal{O}(1)$ be a relatively ample line bundle. Let $\mathcal{E}$ be a coherent sheaf over $X$ and flat over $...

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### Is a cubic hypersurface determined by its Fano variety of lines?

Consider a smooth cubic complex hypersurface $X\subset\mathbf{P}^{n+1}$ of dimension $n\geqslant 3$. The associated Fano variety of lines $F(X)$ is a smooth variety of dimension $2n-4$. Can one ...

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### moduli stack of double covers of $\mathbb{P}^1$ with one marked point

I am trying to improve my moderate knowledge of moduli spaces/stacks by examining the moduli stack of stable double covers of $\mathbb{P}^1$ with one marked point.
My idea is to ignore the stack ...

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345 views

### Isomorphism classes of line bundles with connections

Isomorphism classes of line bundles over a scheme $X$ are described by the Picard group $Pic(X)$. Now there is a paper that describes the moduli space of line bundles with connections. This paper is ...

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### Build a Fuchsian group starting from punctures on a disk

Consider the moduli space of hyperbolic metrics on the disk with $n>3$ marked points on its boundary, $\mathcal{M}_{D,n}$.
$\mathcal{M}_{D,n}$ can be parametrised in terms of cross ratios of the ...

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### Cutting a circle from the hyperbolic plane

Let D be the Poincare' disk its natural hyperbolic metric and with at least 1 marked point on $\partial D$. Suppose I cut an hyperbolic circle of radius $r$ away from it, then I get a Riemann surface ...

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### What is a half cusp in hyperbolic geometry?

I already asked this question on math.stackexchange, but it was suggested that I post it here as well.
The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and ...

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### Intermediate moduli spaces of stable maps

In the following paper:
A. Mustata, M. A. Mustata, "Intermediate moduli spaces of stable maps", Invent. math. 167, 47–90 (2007)
the authors introduced a variation on moduli spaces of stable maps ...

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### Flip of moduli space of stable maps

Let $\overline{M}_{0,2}(G(3,7),4)$ be the moduli space of $2$-pointed degree $4$ stable maps to the Grassmannian of $3$-planes in $\mathbb{P}^7$. Consider the divisor $\Delta$ whose general point is a ...

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### Rim hook decomposition and volume of moduli spaces

I did some computer experiments, counting the number of rim-hook decompositions (aka border-strip decompositions) of rectangles of shape $2n \times n$, where each strip has size $n$.
Here are 12 of ...