All Questions
17 questions
30
votes
3
answers
3k
views
Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches?
The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is an isomorphism in ...
- 21k
22
votes
4
answers
3k
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 ...
- 6,777
18
votes
1
answer
797
views
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 ...
- 19.2k
17
votes
4
answers
2k
views
What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve?
Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well known, cool formula ...
- 445
17
votes
0
answers
553
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 ...
- 2,035
16
votes
1
answer
360
views
Moduli space of boundary maps with prescribed chain and homology groups?
Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every ...
- 15.5k
11
votes
5
answers
3k
views
Ribbon graph decomposition of the moduli space of curves
What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?
- 21k
8
votes
1
answer
529
views
Topology on the space of constructible sheaves
Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
- 21.8k
6
votes
1
answer
675
views
Some questions on the intersection theory on a Hilbert scheme of points of a surface.
If $\Sigma$ is a smooth complex curve in a smooth projective surface $X$, then we can consider the homology class represented by $\Sigma^{[n]} \subset X^{[n]}$. $\ \ $ Where, $X^{[n]}, \Sigma^{[n]}$ ...
- 445
5
votes
1
answer
436
views
Is $M_{1,n}$ affine?
A famous conjecture of Looijenga states that the moduli space of curves $M_{g,n}$ is the union of $g- \delta_{0,n}+ \delta_{0,g}$ open affine subsets, where $g,n$ are non-negative integers satisfying $...
5
votes
1
answer
344
views
Is there a relationship between the moduli space of spatial polygons and the moduli space of labeled points?
It is well known that the set of all polygons with consecutive side lengths $l_1, \dots, l_n$ in $\mathbb{R}^3$, considered up to rigid motions, is a compact complex manifold. Of course, I am assuming,...
- 1,017
5
votes
0
answers
175
views
Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?
In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...
- 641
4
votes
0
answers
98
views
Nakamura graphs and moduli space cellular decomposition
I have recently started studying the cell decomposition of moduli spaces. Among the papers I read, I studied this paper, but there is something I do not understand and I can't find the answer on my ...
- 641
3
votes
0
answers
126
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 ...
- 101
2
votes
1
answer
1k
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, ...
- 2,816
1
vote
1
answer
232
views
Pullback of boundary divisors under forgetful maps
Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n-$pointed smooth rational curve of genus zero and $\overline{\mathbf{U}}_{0,n}$ the universal family described by $\pi_n:\overline{\...
- 31
1
vote
0
answers
120
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 ...
- 31