Pfaffian bundles on cubic surfaces and configurations of planes

TL;DR

This paper establishes a birational correspondence between pfaffian vector bundles on cubic surfaces and configurations of five planes, providing new insights into their geometry and rationality properties.

Contribution

It introduces a canonical birational map linking pfaffian bundles on cubic surfaces with complete pentahedra, and explores applications to advanced geometric structures.

Findings

Established a birational map between moduli space and pentahedra configurations.

Proved a rationality result for the universal case.

Provided explicit normal form for five general lines in P5.

Abstract

We give 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's Hyper-K\"ahler manifolds are also discussed.