Cubic fourfolds with birational Fano varieties of lines

TL;DR

This paper presents examples of cubic fourfolds with Fano varieties of lines that are birationally equivalent, supporting the conjecture that such equivalences imply the fourfolds are also birationally related.

Contribution

It provides new explicit examples of cubic fourfolds with birational Fano varieties of lines and explores their implications for conjectures on rationality and equivalence.

Findings

Examples of non-isomorphic cubic fourfolds with birational Fano varieties of lines.

Evidence supporting the conjecture that Fourier-Mukai partners are birationally equivalent.

Proposed conjecture relating birationality of Fano varieties to that of the fourfolds themselves.

Abstract

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.