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