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25
votes
0answers
454 views

Why do H_4 and M_4 have the same virtual Euler characteristic?

Here's a funny coincidence: The virtual (or "orbifold") Euler characteristic of $\mathcal M_g$ is known by the work of Harer and Zagier: one has $\chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)$. Now ...
1
vote
2answers
421 views

Generalized Quot-schemes

Given $S=\mathbb{P}^2$ and a locally free $O_S$-module $E$ of rank r and an integer $l\geq 1$. Then it is known that the scheme $Quot(E,l)$ is irreducible, due to Ellingsrud and Lehn. Here $Quot(E,l)$ ...
2
votes
0answers
207 views

Projectivized Normal Cone to Satake Compactification

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties over $\mathbb{C}$. There exists a compactification, the Satake compactification, which is minimal and has the ...
8
votes
1answer
296 views

Pullback along the Torelli map is an isomorphism

I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard ...
3
votes
1answer
242 views

notion of stability in a category

This is a question in general sense, but answers about specific examples are also welcome. Why do we need the notion of stability of objects in a category. If we've a subcategory of stable objects, ...
7
votes
1answer
492 views

Normality of a locus of points in projective space

Let $U_{d,n}\subseteq(\mathbb{P}^d)^n$ denote the locus of $n$-distinct points in projective space $\mathbb{P}^d$ that lie on a rational normal curve of degree $d$, and let $V_{d,n}$ denote its ...
13
votes
1answer
487 views

Does a degeneration always have a larger-dimensional automorphism group?

Suppose $\newcommand{\X}{\mathcal{X}}\X$ is an algebraic stack over a field $k$, $\xi$ is a $k$-point which has another $k$-point $x$ in its closure ($x$ is an isotrivial degeneration of $\xi$). ...
5
votes
1answer
549 views

Flat Principal Connections and Homotopy Groups?

I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...
19
votes
1answer
710 views

Conformal blocks vector bundles on $\overline{M}_{g}$ in terms of generalized theta functions?

Conformal field theory uses representation theory to produce various vector bundles on the Deligne-Mumford compactified moduli spaces $\overline{M}_{g}$ and $\overline{M}_{g,n}$, known as bundles of ...
3
votes
1answer
605 views

Basic Questions about Teichmuller's theorem/quadratic differentials

I have some basic questions about Teichmuller's theorem, since I am a beginner, my questions might be very basic. If you can give some hints/answers or cite some references to study from, I will ...
9
votes
2answers
649 views

Are representations of a linearly reductive group discretely parameterized?

Suppose $G$ is a linearly reductive group over a field (say $\mathbb C$). Does somebody know of a proof that any flat family of finite-dimensional representations of $G$ must be locally constant? In ...
6
votes
1answer
468 views

Proposition 3.93 of Harris-Morrison (rational classes on Deligne-Mumford moduli stack vs. rational classes on Deligne-Mumford moduli space)

Proposition 3.93 of Harris-Morrison's "Moduli of Curves" describes the relationship between rational divisor classes on the Deligne-Mumford moduli stack and rational divisor classes on the ...
1
vote
2answers
446 views

Modular interpretation of an action of the linear group SL_2 on the cohomology of an elliptic curve

Let $E$ be an elliptic curve and $x,y \in H^1(E, \mathbb{Q})$ be a basis for the first rational cohomology group of $E$. There is an action of the linear group $SL_2(\mathbb{Q})$ on ...
22
votes
1answer
1k views

How were moduli spaces defined before functors?

People today in algebraic geometry will typically define a moduli space to be the space which represents the functor of families of whatever object they are interested in studying. However, I am ...
10
votes
1answer
531 views

Can a rational family of genus-g curves have generic gonality? Can it be Brill-Noether general?

We know that M_g is general type for g large enough. In particular, the generic genus-g curve is not contained in a (non-isotrivial) rational family parametrized by P^1. In fact, the high-genus ...
11
votes
2answers
566 views

Given a family of curves, when does there exist a fibered surface over Spec Z parametrizing them?

Let $X_p$ be a projective curve over the finite field $\mathbf{F}_p$ (i.e. a projective $\mathbf{F}_p$-scheme pure of dimension 1) for every prime number $p$. Let $X_\mathbf{Q}$ be a projective curve ...
6
votes
3answers
628 views

Upper bounds for ranks of modular jacobians

The following question came to me earlier as a "side question"; something I'd like to know, but which is not totally necessary for what I'm thinking about or doing: Consider the genus 32 curve ...
6
votes
1answer
1k views

Which Riemann surfaces arise from the Riemann existence theorem?

The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi_1(X \backslash \Delta) \to S_d$ ...
12
votes
2answers
444 views

Is there Harer stability for moduli of curves with level structure?

The famous Harer stability theorem asserts that the homology group $H_d(\mathcal{M}_{g,n},\mathbf{Z})$ is independent of g and n in the range $0 \leq 2d < g-1$. This is proven by analyzing the maps ...
4
votes
1answer
347 views

“extend a functor”

Hi, I have probably a basic question. I have a functor $F: Sch \rightarrow Set$, an algebraic stack $M$ with a "universal family" $G\rightarrow M$ and a representability property like this: for every ...
9
votes
4answers
856 views

Why does (Ribbon) Graph (co)Homology Compute (co)Homology of MCG?

The title says it all. I am looking for an explanation or reference for why the homology of the ribbon graph complex computes the cohomology of the mapping class groups of surfaces. I've seen ...
6
votes
1answer
203 views

What is the closure of product loci in A_g?

Let $A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$. For any partition of $g$, one can consider the corresponding locus inside $A_g$ of products of ...
1
vote
1answer
430 views

What are the relative differential forms of a family of (nodal) curves?

What is the definition of relative differential forms of a family $\pi: X \to B$ of (nodal) curves, where $B$ is the base space.
6
votes
1answer
355 views

Choosing tau for elliptic curves over the rational numbers with prescribed ramification data

Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. Let $B$ be the divisor $$B:= \sum [b_i].$$ We consider this data to be fixed. For $d>1$, we define ...
10
votes
1answer
571 views

When can Witten-esque moduli spaces be used to define invariants of geometric structures?

I am trying to understand the big picture around Seiberg-Witten invariants of 4-manifolds. Of course, this points to Taubes work on Gromov-Witten invariants of symplectic manifolds. It is striking ...
12
votes
3answers
594 views

Are there graph models for other moduli spaces?

Recall that a ribbon graph is a graph with a cyclic ordering at each vertex and such that each vertex has valence greater than or equal to 3. This cyclic ordering exactly gives one the information to ...
5
votes
3answers
680 views

Proving that a map is a morphism

Example : Consider the (open, not compactified) moduli space of stable maps $ \mathcal M_g(1,d)$ of maps of smooth curves of genus $ g$ to $\mathbb P^1$. To each map we associate its branch divisor, ...
2
votes
1answer
290 views

Quotient of an abelian surface by a finite group, irreducible components

Given an abelian surface $A$ as the product of two non isogenous elliptic curves $E_1 $ and $E_2$ over $\mathbb{C}$. We also have a smooth 2 dimensional moduli space $M$ of sheaves on $A$ with some ...
5
votes
1answer
369 views

Cohomology of the Moduli of G-bundles on a Curve

For a simple complex group G and Riemann surface X, are the (integral, if possible) cohomology groups of the moduli of holomorphic G-bundles on X written down somewhere, either explicitly or ...
7
votes
1answer
582 views

Reference for the Hodge Bundle

For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to ...
12
votes
5answers
2k views

Deformations of semisimple Lie algebras

In the questions Is "semisimple" a dense condition among Lie algebras? and What is the Zariski closure of the space of semisimple Lie algebras?, something equivalent to the following is ...
2
votes
1answer
475 views

fundamental group of moduli spaces of sheaves

I am considering moduli spaces of sheaves on irreducible holomorphic symplectic manifolds. I haven't seen a general theory to describe the fundamental group of moduli spaces of sheaves yet. Is there ...
8
votes
0answers
272 views

What is the signficance of the existence of a moduli stack to a moduli problem?

This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me. Let's say we begin with a naive moduli problem: we want a moduli space (whatever space ...
12
votes
1answer
377 views

What is the second fundamental form of moduli space?

Away from the hyperelliptic locus, the moduli of curves immerses in the moduli of principally polarized abelian varieties. The ambient space has a riemannian metric, so one can ask about the second ...
6
votes
1answer
610 views

Picard group of $\mathfrak{M}_g$

Let $\mathfrak{M}_g$ denote the moduli stack of Riemann surfaces of genus $g$, it is a smooth complex analytic stack, and is the analytic stack underlying $\mathsf{M}_g$, the moduli stack of complex ...
6
votes
2answers
1k views

Reference request: moduli space of vector bundles

I am trying to study the moduli of holomorphic vector bundles fast and I'm primarily interested to understand: 1) Why and were the stability is important. 2) How are the construction methods. 3) some ...
8
votes
1answer
549 views

Non-representable functor, representable on locally Noetherian schemes?

What is an example of a functor $F : \mathbb{C}\text{-Sch.} \to \text{Sets}$ with the property that the restriction of $F$ to locally Noetherian $\mathbb{C}$-schemes can be represented by a locally ...
5
votes
0answers
210 views

Universal nondegenrate map over the space of complete linear maps

Let $E$ and $F$ be vector spaces and assume for simplicity that $\dim E = \dim F = n$. Recall that the moduli space of complete linear maps from $E$ to $F$ is the space $M(E,F)$ parameterizing all ...
27
votes
12answers
4k 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 ...
2
votes
0answers
653 views

Vector bundles on some non-projective surfaces

Let $X$ be a smooth projective curve over a field $k$ and let $L$ be a line bundle on $X$. I will denote by $S$ the total space of $L$ -- this is a smooth surface over $k$ containing $X$ (as the zero ...
6
votes
1answer
178 views

What's a good dense open of $\bar{M}_g,n(X,\beta)$?

The title says it all, what's a good dense open of $\bar{M}_g,n(X,\beta)$ which play the role of ${M}_g$ in $\bar{M}_g$? My first (naive) guess is maps from a genus $g$ smooth curve to $X$ which ...
2
votes
2answers
607 views

A question in R.C.Penner's paper about Teichmuller space

In R.C.Penner "Decorated Teichmuller theory of boarded surface", on Page 7 and 8, it says that (without proof) the Teichmuller space of surface with $s$ labelled punctures and $r$ labelled boundary ...
2
votes
2answers
259 views

For which composite $N$ does $X_0(N)$ possess a non-cuspidal rational point?

According the the introduction to Mazur's Rational Isogenies of Prime Degree the following question was open in 1978: Let $N$ be one of the integers 39, 65, 91, 125, or 169. Does the modular ...
18
votes
4answers
1k views

Betti numbers of moduli spaces of smooth Riemann surfaces

Where can I find a list of the known Betti numbers of the moduli spaces $\mathcal{M}_{g,n}$ of genus $g$ Riemann surfaces with $n$ marked points? I need it to cross check results by an implemented ...
2
votes
0answers
157 views

How to characterize good “models” of a category

Let ${\bf Cat}$ denote the category of small categories. Recall that for a category $\mathcal{C}$ and a functor $F\colon\mathcal{C}\to{\bf Cat}$, the Grothendieck construction of $F$, which I'll ...
2
votes
1answer
759 views

Question related to the moduli space of Riemann surfaces and a fibration

If $M_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map: $M^1_{g} ...
4
votes
1answer
586 views

Homology dimension of the mapping class group of a surface with boundary

There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is $H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except ...
4
votes
0answers
557 views

Curious propositon in “Les schemas de modules de courbes elliptiques”

Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation): (II ...
5
votes
1answer
508 views

How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?

How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...
1
vote
0answers
288 views

Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is hausdorff

Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is Hausdorff? Note: here $\bar{M_{g,n}}$ is not the Deligne-Mumford space in the usual algebraic ...