Rational Fixed Points for Linear Group Actions

Pietro Corvaja §1 Introduction. A general principle in the theory of diophantine equations asserts that if an equation admits " many " rational solutions, there should be a geometric reason explaining such abundance. We consider here a (multiplicative) semigroup of N ×N matrices with rational entries: we suppose that all of them admit rational eigenvalues and deduce the natural geometrical consequences. Such consequences, stated in Theorem 1.2 below, will concern the algebraic group generated by the given semigroup. Consider the natural action of GL N on N − 1-dimensional projective space P N−1 : for a non-singular matrix with rational entries, the fact of having a rational eigenvalue amounts to having a rational fixed point in P N−1 ; hence we are naturally lead to consider a linear-group action on an arbitrary algebraic variety. We shall suppose that each element of a given Zariski-dense semigroup has rational fixed points and deduce again the natural geometric consequence (Theorem 1.1). More precisely, let κ be a field of characteristic 0, finitely generated over the prime field Q. From now on, by rational we shall mean κ-rational, unless otherwise stated. Let X be an algebraic variety, and G an algebraic group, both defined over κ. Suppose that G acts κ-morphically on X [Bo 2, §1.7]. Our main theorem will be Theorem 1.1. Let the finitely generated field κ, the algebraic group G, the variety X and the action of G on X be as above. Suppose moreover that G is connected. Let Γ ⊂ G(κ) be a Zariski-dense sub semigroup. If the following two conditions are satisfied: (a) for every element γ ∈ Γ there exists a rational point x γ ∈ X(κ) fixed by γ; (b) there exists at least one element g ∈ G with only finitely many fixed points; then (i) there exists a rational map w : G → X, defined over κ, such that for each element g in its domain, g(w(g)) = w(g). If moreover X is projective, then (ii) each element g ∈ G(κ) has a rational fixed point in X(κ). We remark at once that the stronger conclusion that the group G itself admits a fixed point, i.e. the rational map w can be taken to be constant, does not hold in general (see Example 1.8 below). Example 1.8 bis shows that to prove the second conclusion (ii), the hypothesis that the variety X is …