We investigate relations between the properties of an algebra and its varieties of finite-dimensional module structures, on the example of the Jordan plane R=k/ (xy-yx-y^2). Complete description of irreducible components of the representation variety mod (R,n) obtained for any dimension n, it is shown that the variety is equidimensional. The influence of the property of the non-commutative Koszul (Golod-Shafarevich) complex to be a DG-algebra resolution of an algebra (NCCI), on the structure of

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... representation spaces is studied. It is shown that the Jordan plane provides a new example of RCI (representational complete intersection).