Pfaffian bundles on cubic surfaces and configurations of planes

摘要

We construct a canonical birational map between the moduli space of Pfaffian vector bundles on a cubic surface and the space of complete pentahedra inscribed in the cubic surface. The universal situation is also considered and we obtain a rationality result. As a by-product, we provide an explicit normal form for five general lines in P_5. Applications to the geometry of Palatini threefolds and Debarre–Voisin Hyper-K?hler manifolds are also discussed.