The structure of affine algebraic varieties and the Linearization Problem

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We studied the structure of affine algebraic varieties from the viewpoint of the actions of algebraic groups, especially the additive group. We gave a description of the structure of the affine surfaces of some type including the affine pseudo-plane. We show that there exists a close relationship among the big three open problems -- the Linearization Problem, Cancellation Problem, and the Embedding Problem-- through the action of the additive group. Further, we gave a characterization of the affine space of higher dimension in some cases by the algebraic actions.