Corey Brooke’s scientific contributions

We give several examples of pairs of non-isomorphic cubic fourfolds whose Fano varieties of lines are birationally equivalent (and in one example isomorphic). Two of our examples, which are special families of conjecturally irrational cubics in \calC_{12}, provide new evidence for the conjecture that Fourier-Mukai partners are birationally equivalent. We explore how various notions of equivalence for cubic fourfolds are related, and we conjecture that cubic fourfolds with birationally equivalent Fano varieties of lines are themselves birationally equivalent.

We characterize the birational geometry of some hyperk\"ahler fourfolds of Picard rank 3 obtained as the Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls. In each of the two cases considered, we provide a census of the birational models, relating each model to familiar geometric constructions. We also provide structural results about the birational automorphism groups, giving generators in both cases and a full set of relations in one case. Finally, as a byproduct of our analysis, we obtain non-isomorphic cubic fourfolds whose Fano varieties of lines are birationally equivalent.

We show that for a general cubic fourfold X containing a plane P, the Mukai flop of its Fano variety of lines F along the dual plane $P^*$ admits a map to $\mathbb{P}^2$ fibered in geometrically abelian surfaces. The explicit construction of this fibration affords an analysis of the torsor structure of the smooth fibers when the ground field k is not closed.