Special prime Fano fourfolds of degree 10 and index 2

TL;DR

This paper studies special prime Fano fourfolds of degree 10 and index 2, revealing their cohomological properties, period map behavior, and classification of special cases based on a discriminant parameter.

Contribution

It characterizes the cohomology, period map, and existence conditions of special prime Fano fourfolds of degree 10 and index 2, extending Mukai's and Hassett's work.

Findings

Middle cohomology is of K3 type.

Period map is dominant with 4-dimensional fibers.

Classification of special fourfolds based on discriminant d.

Abstract

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.