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If $V_{d, δ}$ denotes the variety of irreducible plane curves of degree $d$ with exactly $δ$ nodes as singularities, Diaz and Harris (1986) have conjectured that $Pic (V_{d, δ})$ is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that $Pic (V_{d, 1})$ is a finite group, so that the conjecture holds for $δ = 1$ . Actually the order of $Pic (V_{d, 1})$ is $6 (d - 2) d^{2} - 3 d + 1)$ , the group being cyclic if $d$ is odd and the product of $ℤ_{2}$ and a cyclic group of order $3 (d - 2) (d^{2} - 3 d + 1)$ if $d$ is even.

Rational fixed points for linear group actions

Pietro Corvaja (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$ , an affine variety $V$ and a finite map $π : V \to G$ , all defined over a finitely generated field $κ$ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $π (V (κ))$ contains a Zariski dense sub-semigroup $Γ \subset G (κ)$ ; namely, there must exist an unramified covering $p : \tilde{G} \to G$ and a map $θ : \tilde{G} \to V$ such that $π \circ θ = p$ . In the case $κ = ℚ$ , $G = 𝔾_{a}$ is the additive group, we reobtain the...

Rational Functions Invariant under a Finite Abelian Group.

H.W. Jr. Lenstra (1974)

Inventiones mathematicae

Rational functions without poles in a compact set

W. Kucharz (2006)

Colloquium Mathematicae

Let X be an irreducible nonsingular complex algebraic set and let K be a compact subset of X. We study algebraic properties of the ring of rational functions on X without poles in K. We give simple necessary conditions for this ring to be a regular ring or a unique factorization domain.

Rational intersection cohomology of projective toric varieties.

Karl-Heinz Fieseler (1991)

Journal für die reine und angewandte Mathematik

Rational intersection cohomology of quotient varieties.

F. Kirwan (1986)

Inventiones mathematicae

Rational intersection cohomology of quotient varieties. II.

F. Kirwan (1987)

Inventiones mathematicae

Rational Isogenies of Prime Degree.

B. Mazur (1978)

Inventiones mathematicae

Rational periodic points for degree two polynomial morphisms on projective space

Benjamin Hutz (2010)

Acta Arithmetica

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 points and Dirichlet series. (Points rationnels et séries de Dirichlet.)

Merel, Loïc (1998)

Documenta Mathematica

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