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3
votes
0answers
83 views

Some questions on Kontsevich's moduli space

Motivation: Work of Eisenbud, Harris, and Mumford shows that $\mathcal M_g$ is of general type when $g≥24$. Moreover, by Logan's function $f(g)$ , $\overline {\mathcal M_{g,n}}$ is of general type for ...
2
votes
0answers
124 views

local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$

Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$. To be more ...
0
votes
0answers
89 views

Generalization of the fiber changing trick for principal bundles?

We know that a principal bundle can induce a fiber bundle as follows: if $F$ is a space which admits a $G$-action then a principal $G$-bundle $p: E \to B$ induces a fiber bundle $p: E \times_G F \to ...
4
votes
1answer
159 views

On push-forward of the constant sheaf for fibrations

Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}_E$ be the constant sheaf on $E$. Let $f_*(\underline{\mathbb{C}}_E)$ denote its direct ...
4
votes
1answer
272 views

Shafarevich conjecture for abelian varieties

In the paper "Arakelov's theorem for abelian varieties" Faltings proves the Shafarevich conjecture for abelian varieties. The statement is the following: Let B be smooth projective a curve, S a ...
1
vote
0answers
87 views

Orders of zeros of section of sheaf

We have a semistable family (fibers has normal crossings and they are reduced, multp 1) $f: X \rightarrow Y,$ of complex curves over a smooth curve $Y.$ The family is smooth over the set ...
-1
votes
1answer
112 views

Construction of fibration over Riemannian Manifold

Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$. More precisely: I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some ...
8
votes
2answers
273 views

Circle Action on Quaternionic Projective Space

Quoting from Wikipedia article on quaternionic projective space: Therefore the quotient manifold $$ \mathbb{HP}^{2}/\mathrm{U}(1) $$ may be taken, writing $U(1)$ for the circle group. It has ...
1
vote
2answers
133 views

Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
2
votes
0answers
98 views

Elliptic fibration arising from a higher genus linear system

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$. Let $L\subset H$ be a ...
1
vote
1answer
150 views

Elliptic fibrations with few singular fibers

It is known that non-isotrivial fibrations of genus $g>0$ curves over the projective line have a bunch of singular fibers. There are at least three of them. It is not difficult to prove that an ...
7
votes
1answer
254 views

Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice ...
3
votes
2answers
380 views

Cup product of cohomology in a Serre spectral sequence

How to use Serre spectral sequence to compute cup product structures? Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...
3
votes
3answers
326 views

Does there exist a holomorphic fibration of genus two over $\mathbb{P}^{1}$ with $7$ nodal singularities?

This is a problem about the holomorphic fibration on a complex manifold. Does there exist a holomorphic fibration of genus two over $\mathbb{CP}^{1}$ with 7 nodal singularities? If you are aware of ...
4
votes
1answer
193 views

Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...
6
votes
2answers
335 views

cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...
5
votes
0answers
163 views

When is a circle fibration a circle bundle?

Let $\pi : E \to B$ be a Serre fibration over a CW complex, with circle fibers. In the orientable case, it is easy to see that $\pi$ is fiber homotopy equivalent to a principal $SO(2)$--bundle. ...
0
votes
0answers
67 views

Characterization of Singular locus

Let A be a complete regular local ring over a field k and B be a complete normal local ring over a field k. We assume that (Krull-dimension of A) > 1. We consider the ring homomorphism f: A ---> B, ...
2
votes
0answers
56 views

cohomology ring of cross-section space of one-point compactification of tangent bundle

Let $M$ be an $m$-manifold whose cohomology is known. Let $TM$ be the tangent bundle of $M$ and $\xi$ be the fibre-wise one-point compactification of $TM$. Then $\xi$ is a $m$-sphere bundle over $M$. ...
4
votes
2answers
346 views

Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers. Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...
3
votes
1answer
128 views

group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281: Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...
3
votes
1answer
227 views

Automorphisms of surfaces

Let $X$ be a projective surface with a morphism $f:X\rightarrow\mathbb{P}^1$. Assume that $f^{-1}(t)\cong\mathbb{P}^1$ for any $t\neq 0$ but $f^{-1}(0)$ is the union of two $\mathbb{P}^1$'s ...
9
votes
2answers
527 views

Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections

I previously asked this on Math.SE but didn't receive a satisfactory answer. Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: ...
2
votes
0answers
148 views

A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$. The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence $$ ...
9
votes
2answers
314 views

Fibrations of projective varieties

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension). Let $g:X\rightarrow ...
3
votes
1answer
266 views

Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial

Let $f:X\longrightarrow B$ be a family of curves, with $f$ relatively minimal, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all ...
2
votes
1answer
149 views

Fibration $p : \tilde Y \to Y$ with discrete fiber induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$

If $X$ is simply connected, locally path connected space and $p : \tilde Y \to Y$ is a covering map then it is easy to show that it induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$. Let's weak ...
2
votes
1answer
186 views

fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$ G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
3
votes
1answer
163 views

What's the geometric statement of this fibrewise integration on a symplectic manifold with Lagrangian fibration?

I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic ...
2
votes
1answer
146 views

Open Books $( \Sigma, \Phi) $ living in Lefschetz Fibrations over the disk $D^2$

I have a question about open books and Lefschetz fibrations over the 2-disk $D^2$. Please let me set it up first, before going on. Setup: Say we have a Lefschetz fibration $f: W^4 \rightarrow D^2 $ ...
6
votes
2answers
556 views

Isotrivial fibrations over $\mathbb P^1$

First of all I want to say that algebraic geometry is not "my field of research" so I apologize if the notation is not standard. $S$ is a smooth complex projective surface with a fibration $f$ over ...
4
votes
1answer
364 views

Classifying space for fibrations with Eilenberg-MacLane space as fibers

The following result seems to be frequently quoted: Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). ...
2
votes
2answers
339 views

A fibration of classifying spaces

Let $G$ be a Lie group, $N$ a closed connected normal subgroup. Let $BG$, $BN$, $B(G/N)$ be the classifying spaces of $G,N$ and $G/N$. Is there a fibration $BN\to BG\to B(G/N)$ ? It seems that such a ...
3
votes
0answers
125 views

Recognizing Simplicial (Quasi)Fibrations

Let's say we are given two finite simplicial complexes, which I will suggestively call $E$ and $B$. We'd like an algorithm for the following decision problem: Does there exist a simplicial map ...
2
votes
1answer
102 views

Going Back-and-Forth Between Different Expressions/“Representations” for Open Books.

I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" ...
4
votes
0answers
284 views

Global geometry of discriminant locus

Let $X$ be a smooth projective threefold, and $\pi : X \to S$ an elliptic fibration over a surface (i.e. flat, with general fiber an elliptic curve). I'm interested in constructing such fibrations ...
3
votes
1answer
143 views

Equivalence of the total spaces of two Serre fibrations with equivalent fibers

Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such ...
2
votes
1answer
145 views

modify a fibration with a fiber of higher multiple

Suppose we have an elliptic fibration $f:X\to \mathbb{P}^1$, with a singular fiber $F$, can we construct an elliptic fibration over $\mathbb{P}^1$ with fiber $nF$?
6
votes
1answer
274 views

Where is simpleness used in the proof of existence of Postnikov towers of principal fibrations?

I've read one proof, rather long, in Allen Hatcher's book. There the key is Lemma 4.70, which uses the relative Hurewicz Theorem. But there is another, shorter proof in J.P.May's book "A concise ...
7
votes
4answers
635 views

What does it mean to speak of a homotopy fibration sequence?

I'm reading a paper in which the following is done. We have a certain particular map of spaces $f:X\to Y$ and then it is said something along the lines of "let $Z_f$ denote the space whose defining ...
8
votes
2answers
477 views

Uniformization of Kodaira fibered surfaces

Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the ...
5
votes
1answer
257 views

$\Pi$, $\Sigma$, and identity types without $\eta$ in comprehension categories

In comprehension categories, dependent sums are defined as a choice of left adjoints for all reindexing functors along display maps, satisfying a Beck-Chevalley condition. Dependent products are ...
34
votes
1answer
766 views

Is an affine fibration over an affine space necessarily trivial?

Let $X$ be an algebraic variety over an alg. closed field with zero char. and let $f:X\to \mathbb{A}^n$ be a smooth surjective morphism, such that all fibers (at closed points) are isomorphic to ...
6
votes
2answers
329 views

Why is the path fibration a strong Hurewicz fibration?

In May and Sigurdsson "Parametrized homotopy theory" there is a general treatment of Hurewicz style model structures in Chapter 4, see definitions 4.2.1 and 4.2.2. I am trying to adapt these to a more ...
0
votes
0answers
116 views

on the fibers over closed points

Let $X$ and $S$ $k$-schemes of finite type . ($k$ a field) and $U$ an open subset of $X$ Let $f:X\rightarrow S$ a $k$-morphism of finite type. We assume that for any closed point $s\in S(\bar{k})$, ...
9
votes
1answer
469 views

What is the difference between internal presheaves and presheaves on a total space?

Suppose that $\mathbb{C}$ is a category with finite limits and that $\mathcal{D}$ is a category internal to $\mathbb{C}$. We can also represent $\mathcal{D}$ as a fibration $\mathbb{D}\to\mathbb{C}$. ...
23
votes
3answers
866 views

Is the counit of geometric realization a Serre fibration?

Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the ...
0
votes
0answers
164 views

Change the fiber of a fibration

Let $F \rightarrow E \rightarrow B$ be a (Serre) fibration of topological spaces. Given a map $F' \rightarrow F$, is there a criterion for the existence or even an explicit construction of a fibration ...
1
vote
1answer
232 views

special Lagrangian n-Torus has Tubular neighbourhood?

Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular neighbourhood which ...
3
votes
1answer
452 views

Free Loops, Moore Paths and the Borel Construction

My question is about the relationship between the free loop space LX of a space X and the (appropriately defined) Borel construction $PX \times_{\Omega X} \Omega X$ which is a homotopy equivalent ...