  We investigate relations between the properties of an algebra and its
varieties of finite-dimensional module structures, on the example of the Jordan
plane $R=k<x,y>/ (xy-yx-y^2)$.
  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 representation spaces is studied. It is shown that the
Jordan plane provides a new example of RCI (representational complete
intersection).
