A special Cayley octad

A Cayley octad is a set of 8 points in P3 which are the base locus of a net of quadrics. Blowing up the points of the octad gives a morphism to P2 defined by the net; the fibres of this morphism are intersections of two quadrics in the net, hence curves of genus 1. The generic fibre therefore has a group structure, and the action of this group on itself extends to a birational action on the whole variety. In particular, if the generic fibre has a large group of rational points, the birational automorphism group, and hence the birational geometry, of the variety must be complicated. It is natural to ask whether the converse is true: if the generic fibre has only a small group of rational points, is the birational geometry of the variety correspondingly simple? In this paper we study a special Cayley octad with the property that the generic fibre has finitely many rational points. In Section 1 we find that such an octad only exists in characteristic 2, and is unique up to projective transformations. Our main results then show that the simplicity of the generic fibre is indeed reected in the simplicity of the birational geometry of our blowup. In Section 2 we show that the cones of nef and movable divisors are rational polyhedral, as predicted by the Morrison{Kawamata conjecture. Finally, in Section 3 we prove that our blowup has the \best possible" birational geometric properties: it is a Mori dream space.