Projective-geometric aspects of homogeneous third-order Hamiltonian operators and applications to WDVV equations

In this talk we will consider third-order homogeneous Hamiltonian operators, introduced by B.A. Dubrovin and S.P. Novikov in 1984. It was recently found that they are in correspondence with quadratic line complexes, which are algebraic varieties in the space of all lines of a complex projective space. This enabled us to classify them in the 2 component case, and, using Segre-Weiler classification of quadratic line complexes, in the 3 component case. Third-order homogeneous operators appear in the formulation of WDVV equations as hydrodynamic-type systems. Thanks to a new set of compatibility conditions between the matrix of velocities of the system and third-order homogeneous Hamiltonian operators we are able to prove existence and uniqueness theorems of such operators for many WDVV systems. The remarkable fact that a quadratic line complex is attached to each WDVV system is maybe related with the underlying projective-geometric nature of the WDVV equations.
Talk is based on joint work with E.V. Ferapontov, M.V. Pavlov.