31. Fixing one such point as a midpoint actually defines an affine structure on the projective line containing that range. 32. First, one can replace the projective line by an " n "-dimensional projective space. 33. One of the Italian mathematicians was Giovanni Vailati who studied the circular order property of the real projective line . 34. The Riemann sphere is one example of a complex algebraic manifold, since it is the complex projective line . 35. The converse of this observation ( except for the projective line ) is the fundamental theorem of projective geometry. 36. I don't understand how the cross ratio answers when 4 points in projective line are projectivly equivalent. 37. Nevanlinna theory addresses the question of the distribution of values of a holomorphic curve in the complex projective line . 38. Many curves, for example hyperelliptic curves, may be presented abstractly, as ramified covers of the projective line . 39. Abstractly we can say that this is a projective line in the space of all conics, on which we take 40. A homography of a projective line may also be properly defined by insisting that the mapping preserves cross-ratios.
