Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

A wonderful compactification of an orbit under the action a semi-simple and simply connected group is smooth projective variety containing as dense open subset, where added boundary divisor simple normal crossing. We construct space symmetric symplectic matrices, investigate its geometry. As application, we describe birational geometry Kontsevich spaces parametrizing conics in Lagrangian Grassmannians.