Building on work of Segre and Kollár on cubic hypersurfaces, we construct over imperfect fields of characteristic $p\geq 3$ particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose rational points are Zariski dense are not necessarily unirational. A likewise behavior holds for certain cubic surfaces in characteristic $p=2$ .

For a hyper-Kähler manifold deformation equivalent to a generalized Kummer manifold, we prove that the action of the automorphism group on the total Betti cohomology group is faithful. This is a sort of generalization of a work of Beauville and a more recent work of Boissière, Nieper-Wisskirchen, and Sarti, concerning the action of the automorphism group of a generalized Kummer manifold on the second cohomology group.

The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups: simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of a certain pentagon in the Leech lattice and also in the complex algebraic geometry of K3 surfaces.

We characterize the Fermat quartic K3 surface, among all K3 surfaces, by means of its finite group symmetries.

It is well known that the so-called Beauville number B = 25.33.52.7.11.13.17.19 is a universal bound of the global indices of ℚ–Calabi–Yau 3-folds, but it has been unknown whether this number is best possible or not.