Hypersurfaces are exactly the subvarieties of dimension of an affine space of dimension of . This is the geometric interpretation of the fact that, in a polynomial ring over a field, the height of an ideal is 1 if and only if the ideal is a principal ideal. In the case of possibly reducible hypersurfaces, this result may be restated as follows: hypersurfaces are exactly the algebraic sets whose all irreducible components have dimension .
