**2**

votes

**0**answers

246 views

### Beautiful curves in Gromov-Witten theory, and in Donaldson-Thomas theory

Let $X$ be a smooth complex projective threefold, and $\beta\in H_2(X,\mathbb Z)$ a curve class. In the Kontsevich's moduli space of stable maps, $\overline M_g(X;\beta)$, a general point $[f:C\to X]$ ...

**5**

votes

**1**answer

161 views

### Reference request: maps between moduli spaces

I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with ...

**2**

votes

**1**answer

132 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 ...

**10**

votes

**1**answer

330 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 ...

**3**

votes

**1**answer

196 views

### Intersection theory on M_{g,n}

Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?

**1**

vote

**0**answers

74 views

### Ampleness of the Canonical Bundle for Siegel Modular Varieties

Background
Throughout I only work with varieties over $\mathbb{C}$.
For $p$ a prime number, Let $Y(p)$ denote the modular curve parametrizing elliptic curves together with full $p$-torsion ...

**3**

votes

**0**answers

350 views

### A quotient stack question

Let $X$ be a proper Deligne-Mumford stack, whose normalization, $X'$, is a global quotient stack (that is, a stack of the form [W/GL_n],where W is an algebraic space) with a projective scheme as a ...

**2**

votes

**0**answers

229 views

### Cech cohomology.

Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with ...

**7**

votes

**2**answers

378 views

### Why do parameter spaces (often) behave like what they parameterize?

From the my limited experience it seems that often a parameter space (miraculously) has some of the properties of the elements it parameterizes. For example:
The parameter space of all plane conics ...

**2**

votes

**0**answers

174 views

### bijection of moduli space of equivariant holomorphic embeddings

Consider the moduli space $\mathcal{M}$ of equivariant holomorphic embeddings of closed oriented Riemann surfaces into a generic quintic three-fold $X$ in $\mathbb{P}^4,$ of given degree $d \in ...

**3**

votes

**1**answer

155 views

### On Universal Abelian surfaces over a Shimura curve.

Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal ...

**2**

votes

**2**answers

251 views

### Moduli space of stable maps into very ample hypersurfaces!

Let $X$ be a smooth complex projective variety and $L$ be some ample divisor.
For a holomorphic map $u:\Sigma \to X$, we define its degree to be $deg(u^*L)$.
Question: For a given positive integer ...

**3**

votes

**1**answer

249 views

### Grothendieck duality for stacks

Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In ...

**1**

vote

**1**answer

219 views

### role of the m-th hilbert point in moduli of curves

I have been learning the construction of $\overline{M}_g$ with GIT. I can identify two parameters: The power of the pluricanonical embedding that defines the map:
$$
C \to \mathbb{P}(H^0(C,K_C^n))
$$
...

**0**

votes

**0**answers

117 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 ...

**4**

votes

**0**answers

216 views

### Deligne-Mumford moduli spaces and compactification of symmetric matrices

The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered ...

**2**

votes

**1**answer

138 views

### stable vector bundle and space surves

I am sure this is well known, but I am not an expert...so I appreciate any help
Let $C \subset \mathbb{P}^3$ be the complete intersection of two hypersurfaces of degree $d_1$ and $d_2$. Let ...

**5**

votes

**1**answer

194 views

### Maps from the moduli space of abelian surfaces with level stucture to curves

Let $A_2(N)$ denote the moduli space of principally polarized abelian surfaces with level $N$ stucture. The absolute Igusa invariants $i_1$ $i_2$ and $i_3$ give three different maps from $A_2(1)$ to ...

**1**

vote

**1**answer

124 views

### divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves

Inside the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}_{0,4}$, and those of ...

**7**

votes

**1**answer

323 views

### Singularities of moduli spaces of curves

Let $\overline{M}_{g,n}$ be the moduli space of $n$-pointed genus $g$ Deligne-Mumford stable curves. This is a normal projective scheme. Then
$$codim_{\overline{M}_{g,n}}Sing(\overline{M}_{g,n})\geq ...

**1**

vote

**2**answers

278 views

### The variety associated to moduli space

Let $M$ be a (fine) moduli space which parametrizes certain varieties (The moduli space in my mind is the moduli space of abelian surfaces with certain polarization -- but I don't know what is the ...

**11**

votes

**0**answers

443 views

### Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...

**2**

votes

**0**answers

136 views

### Are there any results on stable maps to Artin stacks with infinite stabilizers?

The Abramovich-Vistoli/Chen-Ruan theory of twisted stable maps into Deligne-Mumford stacks is extremely useful, as is the generalization to tame Artin stacks in positive characteristic. I am ...

**12**

votes

**0**answers

392 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 ...

**15**

votes

**1**answer

625 views

### Number of curves over a finite field

Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...

**3**

votes

**1**answer

430 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 ...

**4**

votes

**2**answers

205 views

### A question on the existence of the quotient of the Hilbert scheme of tricanonical curves

In order to construct the coarse moduli scheme of smooth projective curves of genus $g$, the classical results of Mumford (using the numerical criterion of stability) say that for large enough $m$, ...

**11**

votes

**1**answer

626 views

### Moduli space of motives vs moduli space of varieties

A (projective) abelian variety $A$ over the complex numbers is determined by $H^1(A,\mathbb{Z})$ together with its Hodge structure and polarization. This miracle means that one can parametrise ...

**1**

vote

**2**answers

343 views

### Connections on the Hodge bundle?

Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of ...

**2**

votes

**2**answers

272 views

### spin bundle vs. hodge bundle

Let $\overline{\mathcal{M}}^r_{g,n}$ the space of $n$ pointed $r$-stable curves of genus $g$, endowed with a root of the canonical sheaf, and let $\mathcal{C} \to \overline{\mathcal{M}}^r_{g,n}$ be ...

**2**

votes

**1**answer

443 views

### ${\rm Ext}^1$ and extensions of line bundles on a curve

I am confused about the following. I know that for two line bundles $L_1, L_2$ on an algebraic curve $C$ the vector space ${\rm Ext}^1(L_1,L_2)$ classifies isomorphism classes of rank two vector ...

**1**

vote

**1**answer

155 views

### Moduli spaces admitting birational morphisms over moduli spaces of curves

There are many alternative compactifications of $M_{g,n}$ which live naturally under the classical Deligne-Mumford compactification $\overline{M}_{g,n}$.
For instace the moduli spaces of weighted ...

**1**

vote

**1**answer

196 views

### Canonical bundle of the moduli space of curves

By the pointed canonical bundle formula the canonical bundle of $\overline{M}_{g,n}$ is given by
$$K_{\overline{M}_{g,n}} = 13\lambda+\psi-2\delta-\sum_{I}\delta_{1,I}$$
where $\lambda$ is the Hodge ...

**1**

vote

**1**answer

281 views

### Kodaira dimension of the moduli space of curves

It is known that the moduli space $\overline{M}_{g}$ of genus $g$ curves is of general type for $g\geq 24$.
By Theorem 2.4 of
Logan, Adam The Kodaira dimension of moduli spaces of curves with ...

**7**

votes

**2**answers

523 views

### $Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space

There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence ...

**1**

vote

**0**answers

147 views

### k-amplitude of the Hodge bundle on $\overline M_g$

Let $\mathbb E$ denote the rank $g$ Hodge bundle on $\overline M_g$, the moduli space of stable curves of genus $g$. It is a nef vector bundle. I wonder what other positivity properties the Hodge ...

**2**

votes

**1**answer

160 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 ...

**2**

votes

**0**answers

156 views

### ampleness in families

Let $X\to S$ be a smooth projective morphism with geometrically connected fibres over an integral noetherian regular scheme $S$ with generic point $\eta$.
Let $L$ be a line bundle on $X$, and suppose ...

**2**

votes

**0**answers

163 views

### branch locus of the discriminant map $\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of pointed, stable, genus $g$ curves. Let $\overline{\mathcal{H}}_{g',r}$ be the hurwitz space of cyclic covers of degree $r$ of genus $g$ curves ...

**5**

votes

**1**answer

478 views

### Tangent space of a Moduli space

Let $X$ be a compact Riemann surface with genus $2$ and $M^2$ the moduli space of stable principal $SL(2)$-bundles of rank $r$. We know that $M^2$ is a complex projective variety of dimention ...

**4**

votes

**0**answers

117 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, ...

**1**

vote

**1**answer

139 views

### Hodge bundle on F-curves

Let $\mathbb{E}\rightarrow\overline{M}_{g,n}$ be the Hodge bundle. Let us cosider an $F$-curve of type $\overline{M}_{1,1}\subseteq\overline{M}_{g,n}$. Is the degree of the restriction of $\mathbb{E}$ ...

**2**

votes

**1**answer

299 views

### Moduli space of stable principal $G$-bundles

We have this Mumford's theorem:
Let $X$ be a Riemann surface of genus $g$, $G$ a simple Lie group. We can consider a principal stable $G$-bundles over $X$ (say $\xi$), where $rk(\xi)=r$ and ...

**4**

votes

**0**answers

139 views

### Shimura varieties and Maximal conditions

Working with Shimura varieties, I have been convinced to call them (or the families giving rise to them especially in $A_{g}$) somehow the "maximal" families. The motivation of this, has been for ...

**2**

votes

**0**answers

243 views

### Points of moduli space of semistable sheaves and S-equivalence classes

Let $X$ be smooth projective and connected curve over an algebraically closed field $k$. One knows the description of $k$-valued points of the moduli space $M_X$ of semistable vector bundles of fixed ...

**1**

vote

**2**answers

352 views

### Is the moduli space of stable vector bundles over a smooth projective curve fano?

Let $K$ be a field of characteristic zero but not algebraically closed. Let $C$ be a smooth projective curve over $K$. Let $r, d$ be two positive integers that are coprime. Consider the moduli space ...

**2**

votes

**1**answer

182 views

### F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$

It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all ...

**1**

vote

**0**answers

165 views

### Moduli space of K3 surfaces and Bogomolov-Tian-Todorov theorem

The famous Bogomolov-Tian-Todorov theorem says that the moduli space of Calabi-Yau manifold is smooth, that is locally a complex manifold. Doesn't this contradicts to the fact that the moduli space ...

**9**

votes

**1**answer

555 views

### There are only finitely many varieties up to deformation

Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert ...

**5**

votes

**0**answers

129 views

### Compactifying the space of indecomposable abelian varieties

Let $A_g$ be the moduli space of principally polarized abelian varieties and $A_g^0$ the open substack of indecomposable ones. Abstractly we know $A_g^0$ has a compactification with complement a ...