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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12
votes
0answers
381 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 ...
14
votes
1answer
590 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
1answer
428 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
2answers
198 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
1answer
568 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
2answers
310 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
2answers
264 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
1answer
437 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
1answer
151 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 ...
0
votes
1answer
177 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
1answer
259 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
2answers
516 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
0answers
139 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
1answer
154 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
0answers
152 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
0answers
154 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
1answer
448 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
0answers
112 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
1answer
136 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
1answer
276 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
0answers
134 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
0answers
234 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
2answers
340 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
1answer
180 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
0answers
147 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
1answer
550 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
0answers
127 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 ...
4
votes
0answers
254 views

Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
4
votes
0answers
58 views

Moduli space of CY3 with $h^{2,1}=1$ is $\mathbb{P}^1\setminus \{p_i\}_i$?

It seems to me that all known CY3 with $h^{2,1}=1$ has the complex moduli space of the form $\mathbb{P}^1\setminus \{p_i\}_i$ for some $i \in \mathbb{N}$. Is this true for all CY3 with $h^{2,1}=1$? ...
2
votes
1answer
194 views

Dimension of the linear system of $\psi$-class on $\bar M_{0;n}$

Consider the (Deligne-Mumford compactification of the) moduli space of complex rational marked curves $\overline M_{0;n}$. For each $i\in \{1,\ldots,n\}$ we can construct a line bundle $L_i$ with a ...
10
votes
0answers
377 views

State of research in moduli space of flat connections

I am a recent PhD student trying to settle into a research topic. Even though I have a current project I am working on, I am not particularly enjoying it and would like to switch. Before braving the ...
1
vote
1answer
277 views

Can the Albanese map be anything?

Sorry for the vague title. This question is about the Albanese map from the variety $M$ of canonically polarized varieties to the set of abelian varieties. (The variety $M$ is not of finite type...) ...
4
votes
0answers
349 views

Possible automorphism groups of a K3 surface

Which finite groups are automorphism groups of polarized K3 surfaces (let's say over ℂ)? ...and does the answer change is I remove "polarized"? (polarized = equipped with an ample line bundle)
2
votes
1answer
341 views

There are many varieties with ample canonical bundle

Let $X$ be a smooth projective connected complex algebraic variety with ample canonical bundle. Let $h$ be the hilbert polynomial of the canonical bundle. Why is the moduli stack of canonically ...
2
votes
1answer
246 views

Rigidification and good moduli space (morphism) in the sense of Alper

Let $\mathcal{X}$ be an Artin stack. In "Abramovich, Graber, Vistoli - Twisted bundles and admissible covers", the authors describe a procedure to rigidify $\mathcal{X}$ by a central subgroup $H$ of ...
8
votes
2answers
492 views

Proving that a generic variety with ample canonical bundle has no automorphisms

Let $X$ be a smooth projective connected variety over the complex numbers with ample canonical bundle. If $X$ is generic and $\dim X \leq1$, the automorphism group of $X$ is trivial, see for instance ...
9
votes
1answer
349 views

examples of “exotic” moduli problems for elliptic curves?

Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as ...
0
votes
0answers
75 views

Analogue of Knudsen clutching

Is there an analogue of Knudsen clutching for moduli stacks of "pointed" varieties? I admit this question is not very precise. I'm really asking two questions though. Are there analogues of the ...
4
votes
0answers
184 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 ...
1
vote
0answers
242 views

Complete curves in $M_g$ and Theta Characteristics

Let $g\geq 3$. Following the reference below, the locus of curves in $M_g$ with an effective even theta characteristic has codimension $1$. (Those are the curves $C$ with an effective line bundle $L$ ...
1
vote
2answers
508 views

are moduli stacks deligne-mumford stacks in general

Let M be your favorite moduli stack over the field of complex numbers. Is it reasonable to expect M to be a Deligne-Mumford stack? I know this is true for the moduli space of curves of genus g, ...
2
votes
2answers
286 views

Is the moduli space of ppAVs smooth?

Let $A_g$ be the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. (EDIT: I mean the coarse moduli space.) Is this smooth? Since $A_g$ is the ...
3
votes
0answers
109 views

state of the art for kodaira dimension of $\overline{\mathcal{M}}_{g,n}$

What is the state of the art about the kodaira dimension (and rationality, unirationality, etc.) of the moduli spaces of $n$-pointed curves of genus $g$? When it is known and when not? It would be ...
1
vote
1answer
122 views

A $\mathbb{Q}$-rational canonical model for $X(N)$?

Before today, every source on the subject talks about algebraic models of the modular curve $X(N)$ over $\mathbb{Q}(\zeta_N)$, but in Ogg's paper "Rational Points on Certain Elliptic Modular Curves", ...
1
vote
1answer
123 views

reference request for the finiteness of cuspidal subgroup of $X_0(N)$?

I've seen stated offhand in many sources that the cuspidal subgroup of the Jacobian of $X_0(N)$ is finite. Do they mean that the subgroup of the jacobian generated by $\mathbb{Q}$-rational cusps is ...
3
votes
1answer
360 views

Is there an elliptic surface over $Y(1)$?

Actually I have a few related questions. Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}_2(\mathbb{Z})\backslash\mathcal{H}$. I know $Y(1)$ is only a coarse moduli space, so there isn't a ...
2
votes
2answers
189 views

smooth modular compactification of moduli of curves

Is there a smooth modular compactification of the moduli space of smooth curves of genus $ g > 1 $ over $ \mathbb{C} $? I am willing to allow for enrichments such as level structures. The ...
1
vote
0answers
220 views

lifts of maps to $\mathcal{M}_{1,1}$

Hi, here's there's a construction about elliptic curves that I do not completely understand. Suppose I consider the two following families of elliptic curves over $\mathbb{C}^*$. The first, which I ...
0
votes
1answer
189 views

universal families and maps to quotient stacks

Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductive group. Roughly ...
5
votes
2answers
454 views

Passage from the moduli functor to the functor of points of the coarse moduli space

Let $F: (Sch)^{o}\to (Set)$ be a functor that admits a coarse moduli $Y$ (a scheme). We can consider $Y$ as a representable functor $h_{Y}: (Sch)^{o}\to (Set)$. Is there a direct way to produce $h_Y$ ...