Abstract. We prove that the moduli space of cubic fourfolds C contains a divisor C 42 whose general member has a unirational parametrization of degree 13. This result follows from a thorough study of the Hilbert scheme of rational scrolls and an explicit construction of examples. We also show that C 42 is uniruled.

We exhibit a Cremona transformation of 4 such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

We prove that if S is a smooth reflexive surface in P 3 defined over a finite field F q , then there exists an F q -line meeting S transversely provided that q ≥ c deg(S), where c = 3+ √ 17 4 ≈ 1.7808. Without the reflexivity hypothesis, we prove the existence of a transverse F q -line for q ≥ deg(S) 2 . √ 17 4 ≈ 1.780776. Recall that a line L meets a surface S transversely if the intersection S ∩ L consists of d = deg(S) distinct geometric points. The reflexivity

We study Cremona transformations that induce bijections on the k-rational points. These form a subgroup inside the Cremona group. When k is a finite field, we study the possible permutations induced on P 2 (k), with special attention to the case of characteristic two. IntroductionWe call a birational self-map of a variety a birational permutation if both it and its inverse are defined on all rational points of the variety. In particular, such a map induces a bijection on the set of rational points. Over a finite field, the rational points form a finite set, so such bijective maps induce permutations in the usual sense. Fixing a variety and a finite ground field, one can ask what kind of permutations on the rational points can be realized this way.