THE FUNDAMENTAL GROUP OF THE GALOIS COVER OF HIRZEBRUCH SURFACE F1(2, 2)

2007 ◽  
Vol 17 (03) ◽  
pp. 507-525 ◽  
Author(s):  
MEIRAV AMRAM ◽  
MINA TEICHER ◽  
UZI VISHNE
Keyword(s):  
Fundamental Group ◽  
Hirzebruch Surface ◽  
Generic Projection ◽  
Galois Cover ◽  
Hirzebruch Surfaces ◽  
Galois Covers ◽  
Branch Curve

This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces Fk(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is [Formula: see text] where c = gcd (a, b) and n = 2ab + kb2. In this paper, we degenerate the Hirzebruch surface F1(2, 2), compute the braid monodromy factorization of the branch curve in ℂ2, and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F1(2, 2) with respect to a generic projection is isomorphic to [Formula: see text].

THE FUNDAMENTAL GROUP OF GALOIS COVER OF THE SURFACE 𝕋 × 𝕋

2008 ◽  
Vol 18 (08) ◽  
pp. 1259-1282 ◽  
Author(s):  
MEIRAV AMRAM ◽  
MINA TEICHER ◽  
UZI VISHNE
Keyword(s):  
Fundamental Group ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Generic Projection ◽  
Galois Cover ◽  
The One ◽  
First Time

This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

scholarly journals Fundamental group of Galois covers of degree 6 surfaces

2021 ◽  
pp. 1-21
Author(s):  
M. Amram ◽  
C. Gong ◽  
U. Sinichkin ◽  
S.-L. Tan ◽  
W.-Y. Xu ◽  
...  
Keyword(s):  
Fundamental Group ◽  
Algebraic Surfaces ◽  
Fundamental Groups ◽  
Galois Cover ◽  
Chern Numbers ◽  
Galois Covers

In this paper, we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations, the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings.

scholarly journals The fundamental group of the complement of the branch curve of the second Hirzebruch surface

Topology ◽  
2009 ◽  
Vol 48 (1) ◽  
pp. 23-40 ◽  
Author(s):  
Meirav Amram ◽  
Michael Friedman ◽  
Mina Teicher
Keyword(s):  
Fundamental Group ◽  
Hirzebruch Surface ◽  
Branch Curve

On Galois covers of Hirzebruch surfaces

1996 ◽  
Vol 305 (1) ◽  
pp. 493-539 ◽  
Author(s):  
B. Moishezon ◽  
A. Robb ◽  
M. Teicher
Keyword(s):  
Hirzebruch Surfaces ◽  
Galois Covers

scholarly journals DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

2019 ◽  
pp. 1-42
Author(s):  
Joachim König ◽  
François Legrand
Keyword(s):  
Hyperelliptic Curves ◽  
Branch Points ◽  
Abc Conjecture ◽  
Galois Cover ◽  
Quadratic Twists ◽  
Large Genus ◽  
Galois Covers ◽  
The One ◽  
Almost All ◽  
Global Principle

We provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ , almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$ . We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$ -rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ . We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of $\mathbb{P}_{\mathbb{Q}}^{1}$ . On the other hand, it allows to generate conditionally ‘many’ curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

scholarly journals Limits of canonical forms on towers of Riemann surfaces

2020 ◽  
Vol 2020 (764) ◽  
pp. 287-304
Author(s):  
Hyungryul Baik ◽  
Farbod Shokrieh ◽  
Chenxi Wu
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Type Theorem ◽  
Universal Cover ◽  
Canonical Forms ◽  
Classical Version ◽  
Galois Cover ◽  
Hyperbolic Form ◽  
Galois Covers ◽  
Hyperbolic Riemann Surface

AbstractWe prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on {S_{n}}’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss–Bonnet-type theorem in the context of arbitrary infinite Galois covers.

The Coxeter Quotient of the Fundamental Group of a Galois Cover of 𝕋 × 𝕋

2006 ◽  
Vol 34 (1) ◽  
pp. 89-106 ◽  
Author(s):  
Meirav Amram ◽  
Mina Teicher ◽  
Uzi Vishne
Keyword(s):  
Fundamental Group ◽  
Galois Cover

scholarly journals Sur l'existence du sch\'ema en groupes fondametal

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Marco Antei ◽  
Michel Emsalem ◽  
Carlo Gasbarri
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Group Scheme ◽  
Final Version ◽  
Finite Fundamental Group ◽  
Galois Covers ◽  
The One

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of \'etale fundamental group. Comment: in French. Final version (finally!)

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