Download PDF by William Fulton: Algebraic Curves: An Introduction to Algebraic Geometry

By William Fulton

ISBN-10: 0201510103

ISBN-13: 9780201510102

ISBN-10: 0805330828

ISBN-13: 9780805330823

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Ai7 + ^26], which imphes a = —f3 and 7 = ^a = —^/3 = —S. Hence c = [02 — ai] is independent of ^. R e m a r k . Existence as well as uniqueness of the intersection point c are obvious, since the configuration lies in the coordinate plane ao V a i V 02 (cf. 4). Viewing this plane affinely, with ai V 02 as the hue at infinity, this fact can be interpreted by saying that the connecting line of points with equal ordinates on the axes are all parallel, which immediately follows from the intercept theorem.

A = a^. On the other hand, each d^ e D obviously belongs toKerF. D Now we ask whether the sequences (ao, . . , a „ ; e) we caUed projective frames actually are frames in the sense of the group-theoretic view onto geometry. According to E. CARTAN [28] this means that the group, on which the geometry is based, acts simply transitively on this set of frames. As will be shown below this does not hold, in general for the group of auto-coUineations; in the important case of the field if = R of real numbers, however, it is true.

Then the foUowing ments are equivalent: a) / is injective; b) D i m / ( P : ) = n ; c) D i m / ( A ) = Dim A for every projective subspace A. state- P r o o f , a) imphes b): Suppose t h a t B := f{P") ^ o and D i m B = k < n. Then there are k + 1 points in general position spanning B: B = 60 V . . V bk- Choose Oj G f^^{bi) ^ o, i = 0,... ,k. Setting A : = OQ V . . V a^ we conclude / ( A ) = B = f{P^) and Dim A < n from (13). Therefore, there is an X G P " \ A for which, nevertheless, f{x) G B has to hold.

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Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton


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