New Techniques in Birational Geometry

Recent work of Galkin and Shinder suggests that if a cubic 4-fold X is rational then the variety F of lines on X is birational to Hilb(K3). I’ll compare this to Hassett’s notion of X having an associated K3 surface, which turns out to be equivalent to saying that F is birational to a moduli space of sheaves on a K3 surface. The key technical tool is the Mukai lattice that Richard Thomas and I introduced a few years ago. I’ll discuss (but not resolve) the question of whether Hassett’s rational cubics containing a plane all satisfy Galkin and Shinder’s condition. Finally I’ll suggest an approach to showing that Hassett’s condition implies Kuznetsov’s derived category condition – always, not just generically – using Lehn’s hyperkaehler 8-fold and Bayer-Macr̀ı.