# Rational fixed points for linear group actions

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

- Volume: 6, Issue: 4, page 561-597
- ISSN: 0391-173X

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topCorvaja, Pietro. "Rational fixed points for linear group actions." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.4 (2007): 561-597. <http://eudml.org/doc/272251>.

@article{Corvaja2007,

abstract = {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 $\pi :V\rightarrow G$, all defined over a finitely generated field $\kappa $ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi (V(\kappa ))$ contains a Zariski dense sub-semigroup $\Gamma \subset G(\kappa )$; namely, there must exist an unramified covering $p:\tilde\{G\}\rightarrow G$ and a map $\theta :\tilde\{G\}\rightarrow V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb \{Q\}$, $G=\mathbb \{G\}_\{a\}$ is the additive group, we reobtain the original Hilbert Irreducibility Theorem. Our proof uses a new diophantine result, due to Ferretti and Zannier [9]. As a first application, we obtain (Theorem 1.1) a necessary condition for the existence of rational fixed points for all the elements of a Zariski-dense sub-semigroup of a linear group acting morphically on an algebraic variety. A second application concerns the characterisation of algebraic subgroups of $\operatorname\{GL\}_N$ admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue.},

author = {Corvaja, Pietro},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

keywords = {Algebraic groups; Hilbert irreducibility},

language = {eng},

number = {4},

pages = {561-597},

publisher = {Scuola Normale Superiore, Pisa},

title = {Rational fixed points for linear group actions},

url = {http://eudml.org/doc/272251},

volume = {6},

year = {2007},

}

TY - JOUR

AU - Corvaja, Pietro

TI - Rational fixed points for linear group actions

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

PY - 2007

PB - Scuola Normale Superiore, Pisa

VL - 6

IS - 4

SP - 561

EP - 597

AB - 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 $\pi :V\rightarrow G$, all defined over a finitely generated field $\kappa $ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi (V(\kappa ))$ contains a Zariski dense sub-semigroup $\Gamma \subset G(\kappa )$; namely, there must exist an unramified covering $p:\tilde{G}\rightarrow G$ and a map $\theta :\tilde{G}\rightarrow V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb {Q}$, $G=\mathbb {G}_{a}$ is the additive group, we reobtain the original Hilbert Irreducibility Theorem. Our proof uses a new diophantine result, due to Ferretti and Zannier [9]. As a first application, we obtain (Theorem 1.1) a necessary condition for the existence of rational fixed points for all the elements of a Zariski-dense sub-semigroup of a linear group acting morphically on an algebraic variety. A second application concerns the characterisation of algebraic subgroups of $\operatorname{GL}_N$ admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue.

LA - eng

KW - Algebraic groups; Hilbert irreducibility

UR - http://eudml.org/doc/272251

ER -

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