Description:

Title: Cubic fourfolds with birational Fano varieties of lines

Abstract: Cubic fourfolds have been classically studied up to birational equivalence, with an eye towards rationality problems. On the other hand, the Fano variety of lines F(X) on a cubic fourfold X is a hyperkahler manifold, and the rationality/irrationality of X is conjecturely reflected in the geometry of the Fano variety of lines. We give examples of conjecturally irrational cubic fourfolds with birationally equivalent Fano varieties of lines. Two of our examples, which are special families in C_12, provide new examples of pairs of cubic fourfolds with equivalent Kuznetsov components. Further, we show the cubic fourfolds themselves are birational. Our examples were discovered by studying the group of birational transformations of the Fano varieties of lines of these cubic fourfolds. This is joint work with Corey Brooke and Sarah Frei, building on our previous work with Xuqiang Qin.