Linearly degenerate PDEs and quadratic line complexes

Abstract

A quadratic line complex is a three-parameter family of lines in projective space P^3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P^3 and taking all lines of the complex passing through p we obtain a quadratic cone with vertex at p. This family of cones supplies P^3 with a conformal structure. With this conformal structure we associate a three-dimensional second order quasilinear wave equation. We show that any PDE arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. This provides a classification of linearly degenerate wave equations into eleven types, labelled by Segre symbols of the associated quadratic complexes. We classify Segre types for which the associated conformal structure is conformally flat, as well as Segre types for which the corresponding PDE is integrable.