  We consider the variety of pre-Lie algebra structures on a given
n-dimensional vector space. The group GL_n(K) acts on it, and we study the
closure of the orbits with respect to the Zariski topology. This leads to the
definition of pre-Lie algebra degenerations. We give fundamental results on
such degenerations, including invariants and necessary degeneration criteria.
We demonstrate the close relationship to Lie algebra degenerations. Finally we
classify all orbit closures in the variety of complex 2-dimensional pre-Lie
algebras.
