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 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 $\textsf{Cr}_2({\textbf{k}})$.

scholarly journals Algebraic Barth-Lefschetz theorems

1996 ◽  
Vol 142 ◽  
pp. 17-38 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Algebraic Variety ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Basic Properties ◽  
Positive Integers ◽  
Lefschetz Theorems ◽  
Algebraic Scheme

We shall work over a fixed algebraically closed field k of arbitrary characteristic. By an algebraic variety over k we shall mean a reduced algebraic scheme over k. Fix a positive integer n and e = (e0, el,…, en) a system of n + 1 weights (i.e. n + 1 positive integers e0, el,…, en). If k[T0, Tl,…, Tn] is the polynomial k-algebra in n + 1 variables, graded by the conditions deg(Ti) = ei i = 0, 1,…, n, denote by Pn(e) = Proj(k[T0, T1,…, Tn]) the n-dimensional weighted projective space over k of weights e. We refer the reader to [3] for the basic properties of weighted projective spaces.

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.

Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces

2002 ◽  
Vol 45 (3) ◽  
pp. 349-354 ◽  
Author(s):  
Marc Coppens
Keyword(s):  
Projective Space ◽  
Linear Systems ◽  
Exceptional Divisor ◽  
Closed Field ◽  
Invertible Sheaf ◽  
Algebraically Closed Field ◽  
Structure Sheaf ◽  
Blowing Up ◽  
Dimensional Projective Space ◽  
Ample Invertible Sheaf

AbstractLet Pn be the n-dimensional projective space over some algebraically closed field k of characteristic 0. For an integer t ≥ 3 consider the invertible sheaf O(t) on Pn (Serre twist of the structure sheaf). Let , the dimension of the space of global sections of O(t), and let k be an integer satisfying 0 < k ≤ N − (2n + 2). Let P1,…,Pk be general points on Pn and let π : X → Pn be the blowing-up of Pn at those points. Let Ei = π−1(Pi) with 1 ≤ i ≤ k be the exceptional divisor. Then M = π*(O(t)) ⊗ OX(−E1 — … — Ek) is a very ample invertible sheaf on X.

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.

scholarly journals Geometric Weil representation in characteristic two

2011 ◽  
Vol 11 (2) ◽  
pp. 221-271 ◽  
Author(s):  
Alain Genestier ◽  
Sergey Lysenko
Keyword(s):  
Weil Representation ◽  
Closed Field ◽  
Triangulated Category ◽  
Algebraically Closed Field ◽  
Witt Vectors ◽  
Suitable Sense ◽  
Characteristic Two ◽  
Greenberg Realization ◽  
Geometric Analogue

AbstractLet k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack Ĝ over k, the metaplectic extension of the Greenberg realization of $\operatorname{\mathbb{S}p}_{2n}(R)$. We also construct a geometric analogue of the Weil representation of Ĝ, this is a triangulated category on which Ĝ acts by functors. This triangulated category and the action are geometric in a suitable sense.

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