Special prime Fano fourfolds of degree 10 and index 2

Laurent Manivel (1) , Olivier Debarre (2) , Atanas Iliev (3)

1 I2M - Institut de Mathématiques de Marseille
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 Department of Mathematics

Mukai proved that most prime Fano fourfolds of degree 10 and index 2 are contained in a Grassmannian G(2,5). They are all unirational and some are rational, as 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 in the period domain, labelled by a positive integer d, the discriminant. We describe special fourfolds for some low values of d. We also characterize those integers d for which special fourfolds do exist.

Domaines: Géométrie algébrique [math.AG]

Laurent Manivel, Olivier Debarre, Atanas Iliev. Special prime Fano fourfolds of degree 10 and index 2. Lecture Notes Series London Mathematical Society, 2015, 417, pp.123-155. ⟨hal-01250867⟩