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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7
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3answers
637 views

The use of Hall algebras in physics

I once read a statement (not memorized precisely) that a certain physics quantity between two states of charge $d_1$ and $d_2$ respectively could be computed by running over the states of charge ...
0
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0answers
115 views

morphisms in the construction of the moduli space of curves by mumford

Hi fellow mathematicians, I'm just studying mumfords proof of the existence of a coarse modulispace for curves of genus $g$in his great GIT book. On page 102 (in the proof of proposition 5.2) he ...
6
votes
2answers
236 views

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 ...
0
votes
1answer
109 views

Regular (or complex analytic) functions on M_3

Let $M_3$ be the moduli space of genus three curves over $\mathbb C$. Are there non-constant regular functions of this space? What about complex analytic functions? This question is prompted by the ...
14
votes
1answer
886 views

Does the moduli space of genus three curves contain a complete genus two curve

Inspired by the question Does the moduli space of smooth curves of genus g contain an elliptic curve and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth ...
2
votes
0answers
156 views

Possible slope for a modular form

Let $f\colon \mathbb{H}_g \to \mathbb{C}$ be a Siegel modular form of weight $k$ with respect to $\Gamma_g$. Then, $f$ admits a Fourier expansion $f(Z) = \sum_T a(T) \exp(i\pi \mathop{tr} TZ)$, where ...
1
vote
0answers
148 views

Moduli Space of an Algebraic K3 surface with singularities.

Suppose that $X$ is an algebraic K3 surface (say polarized). If the singular divisor of $X$ is normal crossing... Do we have a moduli space parametrizing such $K3$ surfaces? If yes do we have a ...
5
votes
1answer
299 views

kapranov's realization of $\overline{M}_{0,n}$ over other fields

Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, or a similar result, ...
5
votes
1answer
308 views

Genus 2 curves vs Abelian surfaces

In the Satake compactification of abelian surfaces we have the following degeneration of a family of abelian surfaces in $\mathbf{H}_2$ $lim_{t \to \infty}\begin{pmatrix} it & b \\\ b & ...
18
votes
2answers
719 views

What's “bad” about unstable sheaves?

To construct a (coarse or fine) moduli space that is separated, one usually throw away some class of the object in question. For moduli of sheaves people talk about (semi-)stability. A coherent sheaf ...
6
votes
1answer
272 views

on a Deformation long exact sequence of moduli space of stable maps

I am reading the book "mirror symmetry" by Hori,Katz,Klemm,etc. And I want to understand the following Deformation long exact sequence \begin{align} 0 & \to Aut(Σ, p_1, . . . , p_n, f)\to Aut(Σ, ...
10
votes
1answer
302 views

What is the Brauer group of the moduli space of (p.p.) abelian varieties?

What is the Brauer group of the moduli space of principally polarized abelian varieties of a given dimension? I am primarily interested in the "open" moduli space, i.e. not a compactification. The ...
4
votes
2answers
403 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
231 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 ...
0
votes
1answer
391 views

base-point free linear system

Let $C$ be a reduced curve in a smooth degree $d$ ($d \ge 5$) surface in $\mathbb{P}^3$. Suppose $C=C_1 \cup ... \cup C_r$ with $C_i$ irreducible and $C_i^2<0$ for all $i$. Then is the linear ...
4
votes
0answers
251 views

Applications of moduli of curves theory

Are there some applications of moduli of curves theory? I was wondering if moduli of curves theory is used (or could be used) for doing research in applied mathematics. I am doing my PhD in algebraic ...
7
votes
3answers
640 views

Families of ideal sheaves: What's the correct definition?

I'm looking at Bridgeland's paper "Flops and Derived categories" and I got confused on what he meant by a family of ideal sheaves. Let $Y$ be a scheme, and let $S$ be another scheme. A family of ...
2
votes
1answer
286 views

Deformation of space curves to union of lines

Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ ...
5
votes
1answer
436 views

Complex structures on a K3 surface as a hyperkähler manifold

A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex ...
0
votes
0answers
142 views

Maximum number of generators of a curve in $\mathbb{P}^3$

Let $H_{d,g}$ denote the Hilbert scheme of curves of degree $d$ and genus $g$ locally of complete intersection in $\mathbb{P}^3$. Given a curve $C \in H_{d,g}$, denote by $S(C)$ a minimal set of ...
8
votes
1answer
541 views

what exactly is the moduli functor for classifying elliptic curves with (full) level N structure?

So, when people say, "the moduli problem of classifying elliptic curves over $\mathbb{C}$ with level $N$ structure", there are usually two associated functors I've seen: $P_N : ...
1
vote
1answer
197 views

When is the natural projection of the HIlbert flag scheme a flat morphism

Let ${Hilb_{P,Q}}_{red}$ be the reduced scheme associated to the Hilbert flag scheme parametrizing all pairs $(C,X)$ with $C \subset X \subset \mathbb{P}^3$, where $C$ is a curve and $X$ a degree $d$ ...
6
votes
2answers
744 views

Floer homology and Invariants for Einstein Field Equations?

Motivation: There have been the instanton (anti-self dual connection) solutions to the Yang-Mills equation $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, leading to the Donaldson ...
5
votes
2answers
565 views

moduli interpretations for modular curves

Some big picture questions - What are some applications of the moduli interpretation for congruence curves? Specifically, the interpretations for congruence curves parametrizing elliptic curves with ...
2
votes
2answers
279 views

General degree $d$ surface in $\mathbb{P}^3$

Let $H_{d_1,g_1}, H_{d_2,g_2}$ be two Hilbert schemes of curves in $\mathbb{P}^3$ with degrees $d_1, d_2$ and genus $g_1, g_2$. Denote by $H:=H_{d_1,g_1}\times H_{d_2,g_2}$ where an element in $H$ is ...
8
votes
1answer
405 views

Mordell-Weil group of the universal abelian scheme

Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$ ...
20
votes
6answers
2k views

Does the moduli space of smooth curves of genus g contain an elliptic curve

Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete. Does $M_g$ contain an elliptic curve? The answer ...
6
votes
1answer
364 views

Deformations of smooth projective hypersurfaces and the Jacobian ring

It is a well-known result of Griffiths that the pieces of Hodge filtration of a smooth hypersurface $X:= (f=0)$ of degree $d$ in $\mathbb{P}^{n}$ are isomorphic to graded pieces of the Jacobian ring ...
5
votes
1answer
317 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
1answer
435 views

Some help in digesting a paragraph in the introduction of Deligne/Rapoport's “Les Schemas de Modules de Courbes Elliptique”

http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as ...
1
vote
1answer
261 views

Fine/Coarse moduli spaces and extensions of fields.

Let $K/k$ be an arbitrary field extension and $X$, $Y$ varieties over $k$ (lets assume projective and perhaps smooth to avoid technicalities). There is a fine moduli space of morphisms between $X$ and ...
1
vote
0answers
149 views

Irreducibility of monodromy of eigenspaces of families of cyclic coverings

In the article "La conjecture de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in ...
3
votes
2answers
402 views

Moduli Space of Abelian Varieties with a N-torsion point

Does there exists (as scheme, or as some sort of stack) the moduli space of principally polarized Abelian Varieties together with a point of order $N$, for $N>1$ an integer? In the case of ...
9
votes
1answer
331 views

Koszulness of the cohomology ring of moduli of stable genus zero curves

Let $n \geq 3$. The ring $H^\bullet(\overline{M}_{0,n},\mathbf Q)$ was determined by Sean Keel. It is generated by the cohomology classes of boundary divisors $D_{A,B}$ corresponding to partitions $A ...
27
votes
1answer
807 views

Is $M_g$ finitely covered by a scheme over the integers?

This question was prompted by my almost-answer to the question Does smooth and proper over $\mathbb Z$ imply rational? , but I never got around to asking this until now. It is well known that $M_g$, ...
2
votes
1answer
227 views

Deformations of pointed stable maps with “curve held rigid” or “preserving the dual graph”

I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order ...
1
vote
0answers
158 views

Smooth curve in the Hilbert flag scheme

Let $d$ be an integer greater than $0$. Let $P_2$ be the Hilbert polynomial of a degree $d$ surface in $\mathbb{P}^3$. Recall, the Hilbert flag scheme $\mathrm{Hilb}_{P_1,P_2}$ parametrizes curves $C$ ...
3
votes
1answer
234 views

Representability of Hom-sheaves of various moduli spaces

(May be a poor title, happy to update) Recall that for a stack $\mathcal{X} \to Sch$ on schemes (e.g. fppf site) and a pair of morphisms $x,y\colon U\to \mathcal{X}$ ($U$ a representable stack) there ...
9
votes
0answers
332 views

gromov witten donaldson thomas correspondence

Let $X$ be a nonsingular projective 3-fold. I am trying to understand the proof of the GW/DT correspondence as presented in Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds. I would ...
0
votes
0answers
210 views

Intersections with divisors on moduli of curves

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of stable curves of genus $g$ with $n$ points. Consider $0 \neq \gamma \in Pic(\overline{\mathcal{M}}_{g,n})$ the first Chern class of a ...
5
votes
1answer
301 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 ...
3
votes
2answers
436 views

Degeneration of projective curves

Given a projective curve $C$, is it possible that $C$ can degenerate into union of lines i.e., does there exist a family of curves $\pi:\mathcal{C} \to B$ such that $\pi^{-1}(0)=C$ and there exists $a ...
1
vote
0answers
143 views

Hurwitz Spaces and Rauch Variational Formulas

I have read in some papers about Rauch-type variational formulas on Hurwitz spaces, and I would like to know what exactly is the theory behind them. A Hurwitz Space $H_g^d$ is the space of coverings ...
3
votes
0answers
247 views

Every curve is a Hurwitz space in infinitely many ways

Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space. A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the ...
3
votes
1answer
410 views

Wanted: differential coming from higher genus surface in Heegaard Floer Homology

I am interested in studying moduli of complex surfaces which arise in computing the differential on the Heegaard Floer Homology chain complex. In particular, I am interested in the generic case, when ...
2
votes
0answers
162 views

Level n-structure as defined by Mumford in GIT

In Mumford's GIT, the definition of level $n$ structure ($n \geq 2)$ is $2g$ sections $\{\sigma_1, \dots, \sigma_{2g}\} : S \rightarrow A$ such that two conditions hold: (i) For geometric points the ...
4
votes
0answers
231 views

Picard group of $M_{g,n}$

It is well known that the Picard group of $M_g$ is $\mathbb{Z}$ generated by the Hodge bundle. What is the Picard group of $M_{g,n}$, the moduli space of curves with marked points?
3
votes
1answer
282 views

How big is the locus of Galois covers in the moduli space of curves

Let's consider the moduli space $M_g$ of curves of genus $g$ over $\mathbf{C}$. Not every curve of genus $g$ is a Galois cover (of the projective line) if $g\geq 3$. How big is the locus of Galois ...
32
votes
2answers
2k views

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 ...
6
votes
0answers
156 views

Relations among Hodge classes?

Let $\pi : C_g \to M_g$ be the universal curve over the moduli stack of genus $g$ curves. Let $\omega_\pi$ be the relative canonical bundle. Then $\mathbb{H} := \pi_\ast \omega_\pi$ is a rank $g$ ...