The algebraic and geometric classification of compatible Pre-Lie algebras

In this paper, we develop a method to obtain the algebraic classification of compatible pre-Lie algebras from the classification of pre-Lie algebras of the same dimension. We use this method to obtain the algebraic classification of complex $2$-dimensional compatible pre-Lie algebras. As a byproduct, we obtain the classification of complex $2$-dimensional compatible commutative associative, compatible associative, and compatible Novikov algebras. In addition, we consider the geometric classification of varieties of cited algebras, that is the description of its irreducible components.