On varieties of Hilbert type

Bary-Soroker, Lior, Fehm, Arno, and Petersen, Sebastian. "On varieties of Hilbert type." Annales de l’institut Fourier 64.5 (2014): 1893-1901. <http://eudml.org/doc/275596>.

@article{Bary2014,
abstract = {A variety $X$ over a field $K$ is of Hilbert type if $X(K)$ is not thin. We prove that if $f\colon X\rightarrow S$ is a dominant morphism of $K$-varieties and both $S$ and all fibers $f^\{-1\}(s)$, $s\in S(K)$, are of Hilbert type, then so is $X$. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.},
affiliation = {Schreiber 208 School of Mathematical Sciences Tel Aviv University Ramat Aviv Tel Aviv 6997801 (Israel); Universität Konstanz Fachbereich Mathematik und Statistik Fach 203 78457 Konstanz (Germany); Fachbereich Mathematik Universität Kassel Heinrich-Plettstr. 40 D-34132 Kassel (Germany)},
author = {Bary-Soroker, Lior, Fehm, Arno, Petersen, Sebastian},
journal = {Annales de l’institut Fourier},
keywords = {Thin set; variety of Hilbert type; Hilbertian field; algebraic group; thin set},
language = {eng},
number = {5},
pages = {1893-1901},
publisher = {Association des Annales de l’institut Fourier},
title = {On varieties of Hilbert type},
url = {http://eudml.org/doc/275596},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Bary-Soroker, Lior
AU - Fehm, Arno
AU - Petersen, Sebastian
TI - On varieties of Hilbert type
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 5
SP - 1893
EP - 1901
AB - A variety $X$ over a field $K$ is of Hilbert type if $X(K)$ is not thin. We prove that if $f\colon X\rightarrow S$ is a dominant morphism of $K$-varieties and both $S$ and all fibers $f^{-1}(s)$, $s\in S(K)$, are of Hilbert type, then so is $X$. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.
LA - eng
KW - Thin set; variety of Hilbert type; Hilbertian field; algebraic group; thin set
UR - http://eudml.org/doc/275596
ER -