41. These arise naturally in projective spaces , though classical irrational rotation on the circle can be adapted too. 42. This means that they cannot be embedded in any projective space as a surface defined by polynomial equations. 43. Every rational variety, including the projective spaces , is rationally connected, but the converse is false. 44. See the article on cohomology for the cohomology of spheres, projective spaces , tori, and surfaces. 45. The dimension of the linear system \ mathfrak { d } is its dimension as a projective space . 46. This result is much more difficult in synthetic geometry ( where projective spaces are defined through axioms ). 47. The real examples can not be converted into the complex case ( projective space over \ C ). 48. Real ( or complex ) finite-dimensional linear, affine and projective spaces are also smooth manifolds. 49. Lines, planes etc . are expanded to the lines, etc . of the complex projective space . 50. But a hyperplane of an " n "-dimensional projective space does not have this property.
