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Rational Isogenies of Prime Degree.

B. Mazur (1978)

Inventiones mathematicae

Rational periodic points for quadratic maps

Jung Kyu Canci (2010)

Annales de l’institut Fourier

Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_{S}$ be the ring of $S$ -integers of $K$ . In the present paper we consider endomorphisms of $ℙ_{1}$ of degree $2$ , defined over $K$ , with good reduction outside $S$ . We prove that there exist only finitely many such endomorphisms, up to conjugation by ${PGL}_{2} (R_{S})$ , admitting a periodic point in $ℙ_{1} (K)$ of order $> 3$ . Also, all but finitely many classes with a periodic point in $ℙ_{1} (K)$ of order $3$ are parametrized by an irreducible curve.

Rational points and Coxeter group actions on the cohomology of toric varieties

Gustav I. Lehrer (2008)

Annales de l’institut Fourier

We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.

Rational smoothness of varieties of representations for quivers of Dynkin type

Philippe Caldero, Ralf Schiffler (2004)

Annales de l’institut Fourier

We study the Zariski closures of orbits of representations of quivers of type $A$ , $D$ ou $E$ . With the help of Lusztig’s canonical base, we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth.

Rationality of the quotient of ℙ2 by finite group of automorphisms over arbitrary field of characteristic zero

Andrey Trepalin (2014)

Open Mathematics

Let $𝕜$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $𝕜$ . Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $𝕜$ is algebraically closed. In this paper we prove that $ℙ_{𝕜}^{2} / G$ is rational for an arbitrary field $𝕜$ of characteristic zero.

Rationally connected varieties over local fields.

Kollár, János (1999)

Annals of Mathematics. Second Series

Rationally Trivial Homogeneous Principal Fibrations of Schemes.

A. Bialynicki-Birula (1970)

Inventiones mathematicae

Reductive group actions on affine varieties and their doubling

Dmitri I. Panyushev (1995)

Annales de l'institut Fourier

We study $G$ -actions of the form $(G : X \times X^{*})$ , where $X^{*}$ is the dual (to $X$ ) $G$ -variety. These actions are called the doubled ones. A geometric interpretation of the complexity of the action $(G : X)$ is given. It is shown that the doubled actions have a number of nice properties, if $X$ is spherical or of complexity one.