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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152 views

Moduli space of flat connections over a torus

Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...
5
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1answer
165 views

Evaluation maps for moduli of stable maps

Let $\overline{M}_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$. Consider the product of the evaluation ...
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97 views

$\mathcal{M}_{g,n}$ a scheme for $n \gg 0$? [duplicate]

I think that for $n \geq 3$, the Deligne-Mumford moduli stack $\mathcal{M}_{0,n}$ is a scheme. Is it more generally true that for every $g$, the Deligne-Mumford moduli stack $\mathcal{M}_{g,n}$ is a ...
7
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198 views

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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0answers
68 views

miniversality vs versality

Consider a moduli problem $\mathcal{M}$. Assume, at each point $x$, the associated deformation problem $\mathcal{M}_x$ has a tangent-obstruction theory. It follows that $\mathcal{M}_x$ has a hull ...
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5answers
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Maryam Mirzakhani's works

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces. Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...
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129 views

Is the moduli space of curves arising from wild ramification smooth?

Fix a natural number $g$, a prime $p$, and a $p$-group $P$. Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...
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65 views

degenerate points in the moduli space of flat principal $G$-bundle with respect to a linear representation on a complex

Let $(A^\bullet,\partial)$ be a complexe of $\mathbb{C}$-vector spaces. We suppose that this complex is of finite length, and all $A^\bullet$ are finite dimensional. Let $H^\bullet$ be the cohomology ...
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1answer
222 views

Proof of “generic curve of genus at least 2 has no nontrivial maps to a positive genus curve”

I searched for it for a long time, but it seems that everybody is taking this for granted and does not bother to point out a proof. Would it be possible that someone points me to a proof or makes me ...
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1answer
320 views

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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637 views

“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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1answer
164 views

Is there a formula for the number of rational cuspidal curves in surfaces other than P^2?

Let $M$ be a two dimensional compact complex manifold and $A \in H_2(M, \mathbb{Z})$ a fixed homology class. Define a rational curve in $M$ to be $\textit{1-cuspidal}$ if the singularities of the ...
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164 views

Quadrics and Moduli Spaces

It is well known that $\overline{M}_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret ...
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1answer
200 views

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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0answers
379 views

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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165 views

projective map from $\overline{\mathcal{M}}_{0,n}$

Suppose I have a morphism $f:\overline{\mathcal{M}}_{0,n} \to \mathbb{P}^N$ birational onto its image, and I know exactly what $F$-curves are contracted (or "dually", what divisors are contracted). ...
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1answer
425 views

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 ...
3
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2answers
302 views

How to flip one triangulation on a surface into another

Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$. Is there an algorithm which takes two ...
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1answer
128 views

What is the value of this hyperelliptic Hodge-type integral?

Consider the moduli space $$ \overline{M}_{0,4}(B\mathbb{Z}/2) $$ This has virtual (and real) dimension one. In a certain sense this moduli space paramaterizes "genus 1 hyperelliptic curves"; that is, ...
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1answer
160 views

Isotriviality: two definitions

Consider a proper flat morphism of $k$-schemes ($k$ is an algebraically closed field) $ f:X\longrightarrow\mathbb P^1_k$ such that every fiber $X_p$ for $p\in\mathbb P^1_{\mathbb C}$ is a reduced ...
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2answers
229 views

sanity check about a morphism from a stack to its coarse moduli space

Let $Y(3)$ be the fine moduli space (say, over $\mathbb{C}$) representing elliptic curves equipped with a full level 3 structure. Abstractly, there are 24 such structures for any elliptic curve, but ...
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4answers
894 views

Details for the action of the braid group B_3 on modular forms

I'm reading Terry Gannon's Moonshine Beyond the Monster, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an ...
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1answer
245 views

Is there a Riemann existence theorem for orbifolds?

For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified ...
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1answer
324 views

Results about moduli of surfaces

There are early successes of the moduli theory - the construction and compactification of the moduli spaces of curves $\overline{\mathcal{M}}_g$ . I want to study about the moduli of algebraic ...
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108 views

Kodaira-Spencer theory in d=1

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$. To determine dimension of $T\mathcal M_g$, start with a complex structure, which in some coordinates can be written ...
6
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2answers
285 views

Moduli spaces of connections as representation spaces

It is well known that the moduli space of flat connections over a closed manifold $M$ can be identified with the representation space $Hom(\pi_1(M), G) / G$. Furthermore, Atiyah and Bott (1983) showed ...
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1answer
178 views

Number of singular fibers in families of hypersurfaces

Consider the projection map $$\pi: X = V(t_0 f + t_1 gh) \to \mathbf P^1,$$ where $[t_0: t_1]$ are the homogeneous coordinates on $\mathbf P^1$, $f=f(x_0, \dots, x_n)$ is a homogeneous polynomial of ...
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0answers
130 views

Properties of finite quotients of quasi-projective varieties

Let $G$ be a finite group acting on a (smooth) quasi-projective variety over $\mathbb C$. One can consider the stacky quotient $[X/G]$ or the "classical" quotient $X/G$. In general, $[X/G]$ is not a ...
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1answer
238 views

Rationality of moduli spaces of rational curves

Let $\overline{M}_{0,n}$ be the moduli space of Deligne-Mumford stable pointed rational curves, and let us consider the quotient $\widetilde{M}_{0,n} = \overline{M}_{0,n}/S_n$. Clearly, there is a ...
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0answers
218 views

On the Hitchin fibration

I will refer to Simpson's "Higgs bundles and local systems". Proposition 1.4: When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
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192 views

Topology of the space of foliations on a 3-manifold

Denote by $\mathcal{P} (M)$ the space of smooth plane fields(oriented and transversely oriented) on a given closed and orientable 3-manifold $M$ with the $C^{\infty}$ topology, and by $\mathcal{F}(M)$ ...
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82 views

About stable maps passing through fixed points

In "Notes On Stable Maps and Quantum Cohomology", Fulton and Pandharipande present some results, and their proofs, about the representability of the functor $\mathcal{M}_{g, n}(X, \beta)$, which maps ...
14
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1answer
398 views

Special fiber of $X(p)$ in characteristic $p$

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_1(p)$ be the fine moduli space representing ...
14
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3answers
461 views

Which mapping class group representations come from algebraic geometry?

Let $\Gamma_g$ be the mapping class group of a closed oriented surface $\Sigma$ of genus $g$. There is a natural surjection $t \colon \Gamma_g \to \mathrm{Sp}(2g,\mathbf Z)$ which sends a mapping ...
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4answers
362 views

Is there a non-abelian version of the Torelli map?

Let $C$ be a connected compact oriented real surface of genus $g$, let $G$ be a connected compact Lie group and let $G_\mathbb{C}$ be the complexification of $G$. One considers the moduli space $M ...
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2answers
305 views

What is the difference between the moduli space of curves and the moduli space of orbi-curves?

Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write. I feel that I should already know the answer to this, but it never sits ...
27
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1answer
815 views

Why is there no Brauer scheme?

Let $X$ be a proper scheme over a base field $k$ (one could consider more general settings, but I am primarly interested in a "geometric" situation with $k$ being algebraically closed). Then the ...
9
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3answers
246 views

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 ...
7
votes
2answers
337 views

Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
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0answers
197 views

Moduli interpretation of Eisenstein series

Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. ...
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0answers
151 views

Moduli interpretation of Hecke operators on Shimura curves

In his book on Automorphic Forms, Shimura gives (chapter 9) definitions of the the Hecke operators for Shimura curves. One can give definitions of the Hecke and Atkin-Lehner operators in terms of the ...
2
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0answers
133 views

Symplectic form on moduli space of connections

Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it. Is there any known construction of a bundle with a ...
4
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143 views

On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces

There are many results on the cohomology of the Hilbert scheme of points of a surface. Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also there are results ...
11
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4answers
798 views

Soft question: beginners reference to moduli spaces

What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces? (Possibly introducing stacks also)? I'm looking for something which really gets the pictures ...
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1answer
153 views

On the generating series of degree $d>1$ Gromov-Witten invariants of the local $\mathbb P^1$

Let $N$ be the total space of the vector bundle $\mathscr O_{\mathbb P^1}(-1)\oplus \mathscr O_{\mathbb P^1}(-1)$ over $\mathbb P^1$, and let $C_0\subset N$ be the zero section. Then $N$ is a ...
2
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1answer
219 views

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 ...
4
votes
1answer
131 views

Canonical bundle of moduli space of rational curves and automorphisms

Let $\overline{M}_{0,n}$ be the usual Deligne-Mumford compactification of $M_{0,n}$ the moduli space of smooth $n$-pointed rational curves. The canonical divisor $K_{\overline{M}_{0,n}}$ can be ...
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0answers
29 views

singular locus of $\mathcal{A}_3(2)^{hyp}$

The geometry of the Satake compactification of $\mathcal{A}_2(2)$ is very well known. It is the singular quartic 3-fold in $P^4$ known as $Igusa\ quartic$. I am looking for references (or ...
4
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2answers
235 views

Non trivial family of hyperelliptic curves

Let $X$ ba a smooth hyperelliptic curve of genus $g$, and let $f:X\rightarrow X$ be the hyperelliptic involution. Consider a $K3$ surface $S$ with an involution $g$ without fixed points. The quotient ...
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13answers
6k views

What is a good introductory text for moduli theory?

Hi,everyone. I am looking for an introductory textbook on moduli theory,about the background on algebraic geometry,I have read Hartshorne chapter1~4. could you please show some good books or roadmap ...