This is a joint work with Alex Massarenti.
A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We will construct the wonderful compactification of the space of symmetric and symplectic matrices on which the symplectic group acts. Next, we will investigate the geometry of this compactification: we will compute the Picard group and we will describe the Mori chamber decomposition of the effective cone in the low-dimensional cases. As an application, we will prove results on the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians.