2
votes
1answer
122 views

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 ...
0
votes
0answers
105 views

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 ...
2
votes
1answer
143 views

picard group of moduli of elliptic r-prym curves

Let $\overline{\mathcal{M}}_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves ...
4
votes
0answers
101 views

Deformations and moduli of semistable sheaves in mixed characteristic

Let $X$ be a projective scheme over an algebraically closed field $k$. There is the coarse moduli space $M_X$ parametrizing semistable sheaves on $X$ with fixed reduced Hilbert polynomial $p$. Now, ...
0
votes
0answers
39 views

Moduli space of stable principal $U(1)$-bundles on a compact Riemann surface

Let $U(1)$ be the simple Lie group defined by $AA^*=1$ and $C$ a compact Riemann surface with genus $2$. How can I describe the moduli space of stable principal $U(1)$-bundles on $C$?
0
votes
0answers
66 views

Moduli space of stable principal $G$-bundles on a compact Riemann surface

Let $C$ be a compact Riemann surface. I'm looking for some references in order to try to understand what is the moduli space of stable principal $G$-bundles on $C$, where $G$ is a simple Lie group. ...
4
votes
0answers
173 views

stability of parabolic bundles beyond the Weyl alcove and Hecke transforms

When considering (quasi)parabolic vector bundles over a smooth complete curve $X$ with marked points $p_i$, it goes back to the foundational work of Seshadri and collaborators that one needs to ...
4
votes
2answers
399 views

What does this quotient of the upper half plane parametrize?

Let $G(N)$ be the congruence subgroup $\big\{ \begin{pmatrix} a&b \\ c&d \end{pmatrix} \in SL_2(\mathbb{Z}) \ \ | \ \ a \equiv d \mod N \textrm{ and } b \equiv c\equiv 0 \mod N \big\}$. ...
4
votes
0answers
223 views

Reference request: virtual fundamental class of moduli of stable maps

Let $f:U\longrightarrow \overline{M}_{0,n}(\mathbb{P}^m,d)$ is the universal family with morphism $\pi:U\longrightarrow\mathbb{P}^m$ and let $X\subset\mathbb{P}^m$ be a hypersurface defined by a ...
5
votes
1answer
299 views

Modular curve parametrizing two cyclic subgroups of an elliptic curve

The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing ...
0
votes
0answers
159 views

What means “extended concepts of symmetry”?

Where could one find a short description oft: "two mathematical extensions of the symmetry - to moduli spaces of sheaves and to derived categories", found here? Happen there interesting things like ...
2
votes
0answers
272 views

Moduli space of points of fixed order N on elliptic curves

Let us consider the moduli space $Y_1(N)$, parametrizing an elliptic curve, together with a choice of a point of N-torsion on it. Over the complex numbers, it is easy to see that this moduli space is ...
3
votes
1answer
402 views

Virtual dimension of moduli of stable maps

I apologize if this question was already asked somwhere else on this website. Let us consider $f:C \to X$ a stable map. This is a point in the moduli space of stable maps. It seems intuitively to me ...
2
votes
2answers
260 views

Reference request: parametrizing covers of the projective line

Hurwitz spaces (or Hurwitz schemes) parametrize covers of the projective line. One can do this in many ways. For example, one could fix the number $r$ of branch points, the degree $n$ of the cover ...
5
votes
2answers
410 views

A convex polyhedral analog of the pentagram map

I am wondering if there is a three-dimensional analog of the pentagram map, which maps a convex polygon to another convex polygon. Here's the Wikipedia image:           ...
8
votes
1answer
300 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 ...
6
votes
1answer
469 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 ...
5
votes
1answer
377 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
617 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 ...
7
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 ...
2
votes
2answers
270 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 ...
1
vote
0answers
290 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 ...
1
vote
3answers
412 views

projective subvarieties of the moduli space of abelian varieties

I know that the fibre of $A_{g,n}$ over $\mathbf{F}_p$ is quasi-projective (of what dimension?). Can one exhibit some smooth projective subvarieties of high dimension in it? What are references for ...
1
vote
3answers
614 views

Are the arithmetic genera of Cohen-Macaulay curves in a fixed homology class bounded?

Let X be a smooth projective variety over the complex numbers. Recall that a Cohen-Macaulay curve is a one-dimensional closed subscheme without embedded or isolated points (fat components are ...
16
votes
3answers
2k views

Conceptual understanding of the Gross-Zagier theorem.

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
14
votes
4answers
916 views

Introductory text for the non-arithmetic moduli of elliptic curves

I'm looking for an introduction to the non-arithmetic aspects of the moduli of elliptic curves. I'd particularly like one that discusses the $H^1$ local system on the moduli space (whether it's $Y(1)$ ...
13
votes
9answers
5k views

Teichmuller Theory introduction

What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?
7
votes
4answers
969 views

cohomology of moduli spaces

Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...