The moduli-spaces tag has no wiki summary.

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### Classification (and automorphisms) of torsion-free modules/sheaves

I would like to know what can be said about the classification of torsion-free modules.
For my purposes, we can assume that $R$ is the function ring of a smooth affine variety over a field. How does ...

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### Open covering of the Hilbert functor of points

Let $R \to A$ be a homomorphism of commutative rings. Define the functor $$\mathrm{Hilb}^n_{A/R} : \mathsf{CAlg}(R) \to \mathsf{Set}$$ as follows: If $B$ is a commutative $R$-algebra, then ...

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### Can Kuranishi families glue together to give a moduli space?

In his article "Moduli spaces of vector bundles on K3 surfaces and symplectic manifolds", Mukai constructs moduli of simple sheaves on a K3 surface through a theorem that asserts the existence of a ...

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### Do principally polarized abelian varieties enjoy a genus expansion?

This is a vague question from an interested outsider:
It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...

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### Kodaira-Spencer theory of deformation done right

I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of ...

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### What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that ...

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### Moduli of curves in characteristic zero

Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}_{g,n}(K)$ and $\overline{M}_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing ...

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### Connected cycles of Shimura curves in $A_{g}$ not contained in larger Shimura subvarieties

Is there always a finite family of Shimura curves $(C_{i})$ in $A_{g}$ the moduli space of principally polarized abelian varieties of dimension $g(\geq 2)$, such that the union $\cup C_{i}$ is ...

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### Arithmetic and moduli spaces of higher genus curves

Modular curves (as moduli of elliptic curves with level structure) play a key role in the study of the arithmetic of elliptic curves. The higher genus curves have very different arithmetic, but I ...

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### Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri.
I quote from the paper-
Can someone please explain how does any non-zero homomorphism ...

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### Number of curves over a finite field

Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...

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### Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ ...

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### Families of Fano varieties over non-hyperbolic curves

Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve.
Let $f:X\to C$ be ...

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### Connectedness of moduli of hypersurfaces

Let $M$ be the coarse moduli space of smooth connected hypersurfaces of degree $d$ in some fixed smooth projective variety $X$.
If $X$ is projective space, it is easy to see that $M$ is connected. ...

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### Moduli space of sheaves on a ribbon

In the paper "A non-linear deformation of the Hitchin dinamycal system", Donagi-Ein-Lazarsfeld describe the irreducible components of the moduli space $\mathcal M_R$ of stable sheaves of numerical ...

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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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### some intuition about the degree of a map

Consider a map
$$ f: \Sigma \to X/\sigma,$$
where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution,
$\sigma:X\to X$ is an antiholomorphic involution
of some ...

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### Characterisation of convergence in Deligne-Mumford compactifiaction

1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex ...

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### Beautiful curves in Gromov-Witten theory, and in Donaldson-Thomas theory

Let $X$ be a smooth complex projective threefold, and $\beta\in H_2(X,\mathbb Z)$ a curve class. In the Kontsevich's moduli space of stable maps, $\overline M_g(X;\beta)$, a general point $[f:C\to X]$ ...

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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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### Grothendieck duality for stacks

Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In ...

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### role of the m-th hilbert point in moduli of curves

I have been learning the construction of $\overline{M}_g$ with GIT. I can identify two parameters: The power of the pluricanonical embedding that defines the map:
$$
C \to \mathbb{P}(H^0(C,K_C^n))
$$
...

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### Cech cohomology.

Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with ...

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### Questions on theorem in Deligne-Mumford's '69 Paper: $\omega_C^n$ is very ample $n\geq 3$

I'm working through the details of Deligne and Mumford's 69' paper, "The Irreducibility of the Space of Curves of Given Genus", and I had a few quick questions:
1) On p. 77, they claim that for $x$ a ...

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### Reference request: maps between moduli spaces

I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with ...

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### Do mapping classes have gonality?

(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.)
The hyperelliptic mapping class group is (by ...

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### Parameter estimation using bayesian update on moduli space?

Scientists take a set of data points, say in ${\mathbb R}^2$, and, assuming that this data should fit a polynomial of degree $d$ (or an exponential, etc.), they estimate parameters.
I would think ...

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### Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?

I'm trying to understand the paper
Arbarello, Enrico, Cornalba, Maurizio,
Calculating cohomology groups of moduli spaces of curves via algebraic geometry.
Inst. Hautes Études Sci. Publ. Math. No. 88 ...

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### Meaning/Origin of Seiberg-Witten Equations/Invariants

Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take an ...

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### Intersection theory on M_{g,n}

Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?

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### Rational curved lying in the boundary of Deligne-Mumford compactification $\bar M_g$

Let $\bar M_g$ be the Deligne-Mumford compactifiction of the moduli space of complex genus $g$ curves $M_g$. Is this correct that through every point of the boundary $\bar M_g\setminus M_g$ passes a ...

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### Teichmuller curves in H(2)

Does anyone know how to prove the infinite union of Teichmuller curves in $\Omega \mathcal{M}_2(2)$ is dense in strata $\Omega \mathcal{M}_2(2)$?
Where $\Omega \mathcal{M}_2(2)$ is the space of ...

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### Ampleness of the Canonical Bundle for Siegel Modular Varieties

Background
Throughout I only work with varieties over $\mathbb{C}$.
For $p$ a prime number, Let $Y(p)$ denote the modular curve parametrizing elliptic curves together with full $p$-torsion ...

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### A quotient stack question

Let $X$ be a proper Deligne-Mumford stack, whose normalization, $X'$, is a global quotient stack (that is, a stack of the form [W/GL_n],where W is an algebraic space) with a projective scheme as a ...

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### The variety associated to moduli space

Let $M$ be a (fine) moduli space which parametrizes certain varieties (The moduli space in my mind is the moduli space of abelian surfaces with certain polarization -- but I don't know what is the ...

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### bijection of moduli space of equivariant holomorphic embeddings

Consider the moduli space $\mathcal{M}$ of equivariant holomorphic embeddings of closed oriented Riemann surfaces into a generic quintic three-fold $X$ in $\mathbb{P}^4,$ of given degree $d \in ...

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### Why do parameter spaces (often) behave like what they parameterize?

From the my limited experience it seems that often a parameter space (miraculously) has some of the properties of the elements it parameterizes. For example:
The parameter space of all plane conics ...

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### On Universal Abelian surfaces over a Shimura curve.

Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal ...

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### Moduli space of genus 2 curves

Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied?

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### Moduli space of stable maps into very ample hypersurfaces!

Let $X$ be a smooth complex projective variety and $L$ be some ample divisor.
For a holomorphic map $u:\Sigma \to X$, we define its degree to be $deg(u^*L)$.
Question: For a given positive integer ...

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### Which one is the correct assertion in the end? (projectivity of moduli of polarized varieties)

I randomly came across a 2007 article by Kollár in which the author makes a mathematical statement and explicitly states that it is in contradiction with a result contained in a 2004 article by ...

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### Deligne-Mumford moduli spaces and compactification of symmetric matrices

The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered ...

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### stable vector bundle and space surves

I am sure this is well known, but I am not an expert...so I appreciate any help
Let $C \subset \mathbb{P}^3$ be the complete intersection of two hypersurfaces of degree $d_1$ and $d_2$. Let ...

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### Maps from the moduli space of abelian surfaces with level stucture to curves

Let $A_2(N)$ denote the moduli space of principally polarized abelian surfaces with level $N$ stucture. The absolute Igusa invariants $i_1$ $i_2$ and $i_3$ give three different maps from $A_2(1)$ to ...

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### divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves

Inside the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}_{0,4}$, and those of ...

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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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### Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...

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### Are there any results on stable maps to Artin stacks with infinite stabilizers?

The Abramovich-Vistoli/Chen-Ruan theory of twisted stable maps into Deligne-Mumford stacks is extremely useful, as is the generalization to tame Artin stacks in positive characteristic. I am ...

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### Lie algebras vs. graph complexes

A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...

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### Action on C[[X,Y]]/f(X,Y) giving complete intersection quotients

Let $R \colon\!= {\Bbb C}[[X,Y]]/(f(X,Y))$ be a complete local ring of Krull-dimension $1$. Assume that we have an action of $\Bbb Z$ on $R$ such that fixed elements by $\Bbb Z$ in $R$ are only ...