scholarly journals Elements Generating a Proper Normal Subgroup of the Cremona Group

2020 ◽  
Author(s):  
Serge Cantat ◽  
Vincent Guirardel ◽  
Anne Lonjou
Keyword(s):  
Normal Subgroup ◽  
Projective Plane ◽  
Infinite Order ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cremona Group ◽  
Birational Transformations ◽  
Proper Normal Subgroup

Abstract Consider an algebraically closed field ${\textbf{k}}$, and let $\textsf{Cr}_2({\textbf{k}})$ be the Cremona group of all birational transformations of the projective plane over ${\textbf{k}}$. We characterize infinite order elements $g\in \textsf{Cr}_2({\textbf{k}})$ having a power $g^n$, $n\neq 0$, generating a proper normal subgroup of $\textsf{Cr}_2({\textbf{k}})$.

Generators of Ideals Defining Certain Surfaces in Projective Space

1996 ◽  
Vol 48 (3) ◽  
pp. 585-595 ◽  
Author(s):  
Sandeep H. Holay
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Ample Divisor ◽  
Plane Curves ◽  
Closed Field ◽  
Divisor Class ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Generating Sets ◽  
Blowing Up

AbstractWe consider the surface obtained from the projective plane by blowing up the points of intersection of two plane curves meeting transversely. We find minimal generating sets of the defining ideals of these surfaces embedded in projective space by the sections of a very ample divisor class. All of the results are proven over an algebraically closed field of arbitrary characteristic.

UNIPOTENT ELEMENTS IN REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE

2012 ◽  
Vol 11 (02) ◽  
pp. 1250038 ◽  
Author(s):  
L. DI MARTINO ◽  
A. E. ZALESSKI
Keyword(s):  
Normal Subgroup ◽  
Irreducible Representation ◽  
Simple Group ◽  
Finite Groups ◽  
Closed Field ◽  
Group Of Lie Type ◽  
Algebraically Closed Field ◽  
Central Quotient ◽  
Representations Of Finite Groups ◽  
Groups Of Lie Type

Let G be a finite quasi-simple group of Lie type of defining characteristic r > 2. Let H = 〈h, G〉 be a group with normal subgroup G, where h is a non-central r-element of H. Let ϕ be an irreducible representation of H non-trivial on G over an algebraically closed field of characteristic ℓ ≠ r. We show that ϕ(h) has at least two distinct eigenvalues of multiplicity greater than 1, unless G is a central quotient of one of the following groups: SL(2, r), SL(2, 9) or Sp(4, 3), and H = G⋅Z(H).

scholarly journals Isomorphisms between complements of projective plane curves

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Mattias Hemmig
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curves ◽  
Complex Numbers ◽  
Closed Field ◽  
Algebraically Closed Field

In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists a line that intersects a unicuspidal curve $C \subset \mathbb{P}^2$ only in its singular point, then any other curve whose complement is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to $C$. This generalizes a result of H. Yoshihara who proved this result over the complex numbers. Moreover, we study properties of multiplicity sequences of irreducible curves that imply that any isomorphism between the complements of these curves extends to an automorphism of $\mathbb{P}^2$. Using these results, we show that two irreducible curves of degree $\leq 7$ have isomorphic complements if and only if they are projectively equivalent. Finally, we describe new examples of irreducible projectively non-equivalent curves of degree $8$ that have isomorphic complements.

scholarly journals Subgroups of elliptic elements of the Cremona group

2021 ◽  
Vol 2021 (770) ◽  
pp. 27-57
Author(s):  
Christian Urech
Keyword(s):  
Projective Plane ◽  
Model Theory ◽  
Complex Projective Plane ◽  
Elementary Model ◽  
Cremona Group ◽  
Birational Transformations ◽  
Derived Length ◽  
Tits Alternative ◽  
Torsion Subgroups ◽  
Complex Projective

Abstract 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 application, 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 Déserti.

Cliques of Irreducible Representations, Quotient Groups, and Brauer's Theorems on Blocks

1995 ◽  
Vol 47 (5) ◽  
pp. 929-945 ◽  
Author(s):  
Harald Ellers
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Quotient Groups

AbstractAssume k is an algebraically closed field of characteristic p and G is a finite group. If P is a p-subgroup of G such that G = PCG(P), and if H is a normal subgroup of G with P ≤ H, then the number of H-cliques of irreducible k[G]-modules is the same as the number of H/P-cliques of irreducible k[G/P]-modules.

ON THE GENERATION OF THE CREMONA GROUP

2020 ◽  
Vol 65 (6) ◽  
pp. 41-45
Author(s):  
Dang Nguyen Dat
Keyword(s):  
Projective Space ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cremona Group

Let k be an algebraically closed field of characteristic 0. We show that any set of generators of the Cremona group Crk(n) of the projective space Pn k with n greater than 2 contains an infinite and uncountable number of non trivial birational isomorphisms.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

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