Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-rational varieties with the Hilbert Property   A variety $X/k$ is said to have the Hilbert Property if $X(k)$ is not thin.
We shall describe some examples of varieties, for which the Hilbert Property is
a new result. We give a criterion for determining when the Hilbert Property for
a variety $X$ implies the Hilbert Property for quotients $X/G$ of the variety
by an action of a finite group. In the case of linear actions of the group $G$,
this gives examples of (non-rational) unirational varieties with the Hilbert
Property, providing positive examples to a conjecture by Colliot-Th\'el\`ene
and Sansuc. We focus then on the study of the Hilbert Property for K3 surfaces
that have two elliptic fibrations, in particular on diagonal quartic surfaces,
i.e. varieties of the form $ax^4+by^4+cz^4+dw^4=0$. We then show, through an
explicit application, how one may use the criterion above to provide other
examples of K3 surfaces with the Hilbert Property. Since the Hilbert Property
is related to an abudance of rational points, K3 surfaces should
(conjecturally) represent a limiting case in dimension 2.

30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph