Geometry of orbit closures for the representations associated to gradings of Lie algebras of types $E_6$, $F_4$ and $G_2$

In this paper we investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on a simple Lie algebras of type $E_6$, $F_4$ and $G_2$. The methods for classifying the orbits for these actions were developed by Vinberg. We give the orbit descriptions, the degeneration partial orders, and decide the normality of the orbit closures. We also investigate the rational singularities, Cohen-Macaulay and Gorenstein properties for the orbit closures. We give the generators of the defining ideals of orbit closures.