63 research outputs found

### Rational fixed points for linear group actions

Let $k$ be a finitely generated field, let $X$ be an algebraic variety and
$G$ a linear algebraic group, both defined over $k$. Suppose $G$ acts on $X$
and every element of a Zariski-dense semigroup $\Gamma \subset G(k)$ has a
rational fixed point in $X(k)$. We then deduce, under some mild technical
assumptions, the existence of a rational map $G\to X$, defined over $k$,
sending each element $g\in G$ to a fixed point for $g$. The proof makes use of
a recent result of Ferretti and Zannier on diophantine equations involving
linear recurrences. As a by-product of the proof, we obtain a version of the
classical Hilbert Irreducibility Theorem valid for linear algebraic groups.Comment: 35 pages, Plain Tex. A gap in the previous proof of Theorem 1.2
overcome, plus minor changes. Thanks to J. Bernik and the refere

### On the Hilbert Property and the Fundamental Group of Algebraic Varieties

We review, under a perspective which appears different from previous ones,
the so-called Hilbert Property (HP) for an algebraic variety (over a number
field); this is linked to Hilbert's Irreducibility Theorem and has important
implications, for instance towards the Inverse Galois Problem.
We shall observe that the HP is in a sense `opposite' to the Chevalley-Weil
Theorem, which concerns unramified covers; this link shall immediately entail
the result that the HP can possibly hold only for simply connected varieties
(in the appropriate sense). In turn, this leads to new counterexamples to the
HP, involving Enriques surfaces. We also prove the HP for a K3 surface related
to the above Enriques surface, providing what appears to be the first example
of a non-rational variety for which the HP can be proved.
We also formulate some general conjectures relating the HP with the topology
of algebraic varieties.Comment: 24 page

### A lower bound for the height of a rational function at $S$-unit points

Let $\Gamma$ be a finitely generated subgroup of the multiplicative group
\G_m^2(\bar{Q}). Let p(X,Y),q(X,Y)\in\bat{Q} be two coprime polynomials not
both vanishing at $(0,0)$; let $\epsilon>0$. We prove that, for all
$(u,v)\in\Gamma$ outside a proper Zariski closed subset of $G_m^2$, the height
of $p(u,v)/q(u,v)$ verifies $h(p(u,v)/q(u,v))>h(1:p(u,v):q(u,v))-\epsilon
\max(h(uu),h(v))$. As a consequence, we deduce upper bounds for (a generalized
notion of) the g.c.d. of $u-1,v-1$ for $u,v$ running over $\Gamma$.Comment: Plain TeX 18 pages. Version 2; minor changes. To appear on
Monatshefte fuer Mathemati

### Integral points, divisibility between values of polynomials and entire curves on surfaces

We prove some new degeneracy results for integral points and entire curves on
surfaces; in particular, we provide the first example, to our knowledge, of a
simply connected smooth variety whose sets of integral points are never
Zariski-dense (and no entire curve has Zariski-dense image). Some of our
results are connected with divisibility problems, i.e. the problem of
describing the integral points in the plane where the values of some given
polynomials in two variables divide the values of other given polynomials.Comment: minor changes, two references adde

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