We analyze (complex) prime Fano fourfolds of degree 10 and index 2. Mukai gave in [M1] a complete geometric description; in particular, most of them are contained in a Grassmannian G(2, 5). As in the case of cubic fourfolds, they are unirational and some are rational, as already remarked by Roth in 1949.
We show that their middle cohomology is of K3 type and that their period map is dominant, with smooth 4-dimensional fibers, onto a 20-dimensional bounded symmetric period domain of type IV. Following Hassett, we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian G(2, 5). Special fourfolds correspond to a countable union of hypersurfaces (the Noether-Lefschetz locus) in the period domain, labelled by a positive integer d. We describe special fourfolds for some low values of d. We also characterize those integers d for which special fourfolds do exist.
Dedicated to Robert Lazarsfeld on the occasion of his sixtieth birthday
One of the most vexing classical questions in complex algebraic geometry is whether there exist irrational smooth cubic hypersurfaces in P5. They are all unirational, and rational examples are easy to construct (such as Pfaffian cubic fourfolds) but no smooth cubic fourfold has yet been proven to be irrational. The general feeling seems to be that the question should have an affirmative answer but, despite numerous attempts, it is still open.
In a couple of very interesting articles on cubic fourfolds ([H1], [H2]), Hassett adopted a Hodge-theoretic approach and, using the period map (proven to be injective by Voisin in [V]) and the geometry of the period domain, a 20-dimensional bounded symmetric domain of type IV, he related geometric properties of a cubic fourfold to arithmetical properties of its period point.
We do not solve the rationality question in this paper, but investigate instead similar questions for a nother family of Fano fourfolds (see Section 2 for their definition).