Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEmbedding problems and open subgroups
148
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the properties of the fundamental group of an affine curve over an algebraically closed field of characteristic $p$, from the point of view of embedding problems. In characteristic zero, the fundamental group is free, but in characteristic $p$ it is not even $omega$-free. In this paper we show that it is almost $omega$-free, in the sense that each finite embedding problem has a proper solution when restricted to some open subgroup. We also prove that embedding problems can always be properly solved over the given curve if suitably many additional branch points are allowed, in locations that can be specified arbitrarily; this strengthens a result of the first author.
rate research
Read More
The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an appliction, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Deserti.
We survey some principal results and open problems related to colorings of algebraic and geometric objects endowed with symmetries.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C/k$ be a smooth connected affine curve. Denote by $pi_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed normal subgroups $N$ of $pi_1(C)$. In Normal subgroups of fundamental groups of affine curves in positive characteristic we proved the same result under the additional hypothesis that $k$ had countable cardinality.
The Poincare polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the $H$-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more generally, to any algebraic variety $X$ where there is a connected, solvable, algebraic group acting with a finite number of orbits. We illustrate this situation by describing the $H$-polynomials of certain projective $Gtimes G$-varieties $X$, where $G$ is a semisimple group and $B$ is a Borel subgroup of $G$. This description is made possible by finding an appropriate cellular decomposition for $X$ and then describing the cells combinatorially in terms of the underlying monoid of $Btimes B$-orbits. The most familiar example here is the wonderful compactification of a semisimple group of adjoint type.
Let $pi_1(C)$ be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic $p>0$ of countable cardinality. Let $N$ be a normal (resp. characteristic) subgroup of $pi_1(C)$. Under the hypothesis that the quotient $pi_1(C)/N$ admits an infinitely generated Sylow $p$-subgroup, we prove that $N$ is indeed isomorphic to a normal (resp. characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of $N$ is a free profinite group of countable cardinality.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
