Quartic curves in characteristic 2

1995 ◽  
Vol 117 (3) ◽  
pp. 393-414 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Finite Deformation ◽  
Normal Forms ◽  
Positive Characteristic ◽  
Surface Singularities ◽  
The Subject ◽  
Characteristic 2 ◽  
Simple Singularities ◽  
Module Type ◽  
Quartic Curves ◽  
Article 5

Simple singularities in positive characteristicSimple singularities in positive characteristic have been discussed by many authors, and the article [5] in particular establishes the subject on a firm footing. In it a simple, or ‘ADE’ singularity is defined by a list of normal forms and it is shown that the following conditions on a singularity are equivalent: (i) it is simple, (ii) it has finite deformation type, (iii) it has finite Cohen-Macaulay module type. Moreover, the normal forms for surface singularities coincide with the earlier list of Artin [1] and those for curves with the list of [9]: in those papers further characterizations were obtained.

scholarly journals Atuação das organizações sociais na administração dos serviços de saúde: eficiência e seu instrumento de concretização do direito fundamental a saúde - Action of social organizations in the administration of health services

2019 ◽  
Vol 1 (2) ◽  
Author(s):  
Laura Garbini Both ◽  
André Rodrigues Meneses
Keyword(s):  
Social Services ◽  
Public Interest ◽  
Economic Sector ◽  
Public Authorities ◽  
Private Partnership ◽  
The Public ◽  
Social Organizations ◽  
The Subject ◽  
Article 5 ◽  
Percentage Data

<p>O presente trabalho objetiva analisar a atuação, legalidade e eficiência das organizações sociais. Uma vez que, esta tem sido motivo de intensos questionamentos, por parte daqueles que não enxergam benefícios na criação de um terceiro setor econômico. Há quem defenda que, é dever exclusivo do poder público, executar e fiscalizar os serviços sociais. A contrário senso há quem defenda uma publicização dos serviços que não são executados apenas pelo poder estatal, mas também pelo setor privado. Sendo assim, porque contrariar uma parceria publico-privada que só objetiva trazer benefícios para a população brasileira?</p><p>No decorrer deste estudo, será respondido tal questionamento, por meio de reflexões acerca das discussões e alegações de inconstitucionalidade da lei 9.637/98, de parte da lei de licitações ─ 8.666/93. Bem como, da suposta violação dos seguintes preceitos constitucionais: artigo 5ª, XVII e XVIII; artigo 22, XXVII; artigo 23; artigo 37, II, X e XXI; artigo 40, caput e § 4º; artigos 70, 71 e 74; artigo 129; artigo 169; artigo 175; artigo 196; artigo 197; artigo 199, § 1º; artigo 205; artigo 206; artigo 208; artigo 209; artigo 215; artigo 216, § 1º; artigo 218 e artigo 225. Onde será comprovado por meio de dados percentuais a eficiência e os benefícios advindos da sua criação.</p><p> </p><p> </p><p> </p><p>This paper aims to analyze the performance, legality and efficiency of social organizations. Since this has been the subject of intense questions from those who do not see benefits in the creation of a third economic sector. There are those who argue that it is the exclusive responsibility of the public authorities to execute and supervise social services. On the contrary, there are those who advocate an advertisement of services that are not only carried out by state power, but also by the private sector. So, why oppose a public-private partnership that only aims to bring benefits to the Brazilian population?</p><p>In the course of this study, this question will be answered, through reflections on the discussions and allegations of unconstitutionality of Law 9.637 / 98, part of the law of bidding - 8.666 / 93. As well as the alleged violation of the following constitutional precepts: Article 5, XVII and XVIII; article 22, XXVII; Article 23; Article 37, II, X and XXI; article 40, caput and paragraph 4; Articles 70, 71 and 74; article 129; Article 169; article 175; Article 196; article 197; article 199, paragraph 1; Article 205; Article 206; article 208; Article 209; Article 215; article 216, paragraph 1; article 218 and article 225. Where will be proven by means of percentage data the efficiency and the benefits coming from its creation.mptions that justify the use of them with greater efficiency in the achievement of the public interest.</p>

scholarly journals Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

Maximal T-spaces of free associative algebras over a finite field

2018 ◽  
Vol 17 (04) ◽  
pp. 1850064
Author(s):  
C. Bekh-Ochir ◽  
S. A. Rankin
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Associative Algebras ◽  
Prime Field ◽  
Characteristic 2 ◽  
Infinite Set

In earlier work, it was established that for any finite field [Formula: see text] and any nonempty set [Formula: see text], the free associative (nonunitary) [Formula: see text]-algebra on [Formula: see text], denoted by [Formula: see text], had infinitely many maximal [Formula: see text]-spaces, but exactly two maximal [Formula: see text]-ideals (each of which was shown to be a maximal [Formula: see text]-space). This raises the interesting question as to whether or not the maximal [Formula: see text]-spaces can be classified. However, aside from the two maximal [Formula: see text]-ideals, no examples of maximal [Formula: see text]-spaces of [Formula: see text] have been identified to this point. This paper presents, for each finite field [Formula: see text], an infinite set of proper [Formula: see text]-spaces [Formula: see text] of [Formula: see text], none of which is a [Formula: see text]-ideal. It is proven that for any distinct integers [Formula: see text], [Formula: see text]. Furthermore, it is proven that for the prime field [Formula: see text], [Formula: see text] any prime, [Formula: see text] is a maximal [Formula: see text]-space of [Formula: see text]. We conjecture that for any finite field [Formula: see text] of positive characteristic different from 2 and each integer [Formula: see text], [Formula: see text] is a maximal [Formula: see text]-space of [Formula: see text]. In characteristic 2, the situation is slightly different and we provide different candidates for maximal [Formula: see text]-spaces.

scholarly journals Spherical subgroups in simple algebraic groups

2015 ◽  
Vol 151 (7) ◽  
pp. 1288-1308
Author(s):  
Friedrich Knop ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Closed Subgroup ◽  
Positive Characteristic ◽  
Dense Orbit ◽  
Simple Algebraic Group ◽  
Spherical Subgroup ◽  
Group A ◽  
General Deformation ◽  
Characteristic 2

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is said to be spherical if it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

Part 2 Jurisdiction, Admissibility, and Applicable Law: Compétence, Recevabilité, Et Droit Applicable, Art.5 Crimes within the jurisdiction of the Court/Crimes relevant de la compétence de la Cour

2016 ◽  
Author(s):  
Schabas William A
Keyword(s):  
International Law ◽  
Subject Matter ◽  
Criminal Court ◽  
Customary Law ◽  
Crimes Against Humanity ◽  
Judicial Interpretation ◽  
Applicable Law ◽  
Preparatory Committee ◽  
The Subject ◽  
Article 5

This chapter comments on Article 5 of the Rome Statute of the International Criminal Court. Article 5 sets out the subject-matter jurisdiction of the Court. It declares that the jurisdiction is limited to ‘the most serious crimes of concern to the international community as a whole’. It lists the four crimes over which the Court has subject-matter jurisdiction: (i) the crime of genocide; (ii) crimes against humanity; (iii) war crimes; and (iv) the crime of aggression. The chapter argues that the function of Article 5 seems largely symbolic, a consequence of the drafting history. At its beginnings, when it was article 22 of the International Law Association 1993 draft, article 5 was described as the ‘core’ or the ‘heart’ of the Court's jurisdiction ratione materiae, providing an enumeration of crimes whose detailed description was to be left to treaties, customary law, and judicial interpretation. But the Preparatory Committee insisted upon precise definitions, and as the texts emerged — they became articles 6, 7, and 8 of the Statute — the function of article 5 became increasingly redundant.

Abstract Witt Rings When Certain Binary Forms Represent Exactly Four Elements

1993 ◽  
Vol 45 (6) ◽  
pp. 1184-1199 ◽  
Author(s):  
Craig M. Cordes
Keyword(s):  
Group Ring ◽  
Positive Characteristic ◽  
Finitely Generated ◽  
Binary Forms ◽  
Witt Ring ◽  
Witt Rings ◽  
Elementary Type ◽  
Characteristic 2

AbstractAn abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.

scholarly journals Fields with few types

2013 ◽  
Vol 78 (1) ◽  
pp. 72-84
Author(s):  
Cédric Milliet
Keyword(s):  
Division Ring ◽  
Positive Characteristic ◽  
Field Extension ◽  
Order Structure ◽  
Small Field ◽  
Finite Degree ◽  
Locally Finite ◽  
First Order ◽  
Finite Dimensional ◽  
Characteristic 2

AbstractAccording to Belegradek, a first order structure is weakly small if there are countably many 1-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic 2 is a field.

Weierstrass points of order three on smooth quartic curves

2019 ◽  
Vol 18 (01) ◽  
pp. 1950020 ◽  
Author(s):  
Alwaleed Kamel ◽  
Waleed Khaled Elshareef
Keyword(s):  
Projective Plane ◽  
Weierstrass Points ◽  
Smooth Projective Plane ◽  
Quartic Curve ◽  
The Subject ◽  
Quartic Curves

In this paper, we study the [Formula: see text]-Weierstrass points on smooth projective plane quartic curves and investigate their geometry. Moreover, we use a technique to determine in a very precise way the distribution of such points on any smooth projective plane quartic curve. We also give a variety of examples that illustrate and enrich the subject.

XII.—Some Remarks occasioned by the Geometry of the Veronese Surface

1942 ◽  
Vol 61 (2) ◽  
pp. 140-159
Author(s):  
W. L. Edge
Keyword(s):  
Subject Matter ◽  
Algebraic Representation ◽  
Fresh Properties ◽  
Veronese Surface ◽  
Quartic Curve ◽  
Mise En Scene ◽  
Dual Character ◽  
The Subject ◽  
Quartic Curves

The subject-matter of these pages may be briefly summarised as follows: the geometry of the Veronese surface, with an algebraic representation of it that does justice to its self-dual character; the relations of the secant planes of the surface to quadrics which either contain the surface or are outpolar to it; and the derivation of an invariant and two contravariants of a ternary quartic in the light of the (1, 1) correspondence between the quartic curves in a plane and the quadrics outpolar to a Veronese surface. There is no suggestion of discovering fresh properties of the surface, though possibly the results in § 12 § 13 may be new; but the geometrical considerations lead naturally to some algebraical results which it seems worth while to have on record, such as, for example, the identity 8.2 and the remarks concerning the rank of the determinant which appears there, and the form found in § 13 for the harmonic envelope of a plane quartic curve. These algebraical results lie very close to properties of the surface; so close in fact that one might say that the Veronese surface is the proper mise en scène for them.

scholarly journals Local-global principles for Weil–Châtelet divisibility in positive characteristic

2017 ◽  
Vol 163 (2) ◽  
pp. 357-367 ◽  
Author(s):  
BRENDAN CREUTZ ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curve ◽  
Rational Point ◽  
Positive Characteristic ◽  
Number Fields ◽  
Global Field ◽  
Global Fields ◽  
Characteristic 2 ◽  
Global Principles ◽  
Global Principle

AbstractWe extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

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