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 SS-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