Birational geometry of Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls

TL;DR

This paper explores the birational geometry of hyperk"ahler fourfolds derived from Fano varieties of lines on special cubic fourfolds, identifying models, automorphisms, and birational relations.

Contribution

It provides a detailed classification of birational models and automorphism groups for these fourfolds, revealing new relations and non-isomorphic examples.

Findings

Identified all birational models related to the Fano varieties.

Explicit birational maps between different models.

Discovered non-isomorphic cubic fourfolds with birationally equivalent Fano varieties.

Abstract

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 identify all of the birational models, relating each model to familiar geometric constructions, and give explicit birational maps between them. 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.