Abstract. Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we extend some earlier work of Manin on this subject. We then focus on the case where k is a finite field, where we show that all except possibly three explicit del Pezzo surfaces of degree two are unirational over k.

We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.

We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface X whose transcendental lattice is isometric to 6 ⊕ 2 .

We consider an elliptic surface π : Ᏹ → ‫ސ‬ 1 defined over a number field k and study the problem of comparing the rank of the special fibers over k with that of the generic fiber over k(‫ސ‬ 1 ). We prove, for a large class of rational elliptic surfaces, the existence of infinitely many fibers with rank at least equal to the generic rank plus two.