Algebraic Geometry 3: Further Study of Schemes (Translations - download pdf or read online

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By Ueno K., Kato G.

This is often the 3rd and ultimate a part of a textbook meant for a graduate path on algebraic geometry (the first have been released as volumes 185 and 197 of this series). Containing chapters 7 via nine, in addition to the suggestions to workouts, the writer covers the elemental houses of scheme conception, algebraic curves and Jacobian kinds, analytic geometry, and Kodaira's vanishing theorem. Translated from the japanese Daisu kika.

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51) As a useful exercise, we suggest that the reader carry out these calculations for the special case f(z) = ez. He will obtain 37 38 E N T I R E FUNCTIONS Of c o u r s e , this r e s u l t can be r e p r e s e n t e d in the form ,- = ^ = 4 . - ! J. If we set z = a in (51) and note that, by hypothesis,/(a) = 0, we obtain Thus, if a is a zero of the function f(z), the constant t e r m in the expansion (51) is equal to z e r o . It may happen that the c o ­ efficients of some of the t e r m s following c 0 are also equal to zero (for example, we may have q =

Anzn + ... (59) as a sort of polynomial of infinitely high degree. We are now at a stage where we can check the soundness of that point of view. If the analogy is valid, the equation "of infinitely high degree'' a0 + fllz-{-... + aHz*-\-... =0 (60) must have infinitely many roots. However, we are immediately disappointed in this. The equation 1 +T -T + # + - + -S- + - = 0 ' <61> which is simply the equation e* = 09 does not have any root at all, as was shown in Section 7. But the situation can be saved by a slight though valuable compromise.

This contradic­ tion proves the theorem. In particular, we may assert that ez, cos z, and sin z, are transcendental functions. 17. " In the first place, in the series f{z) = a0 + alz + aiz* + ... +anz*+ ... , we encounter terms of arbitrarily high powers of z with nonzero coefficients. In the second place, the maximum absolute value M (r; /) of such a function increases more rapidly than does the THE MAXIMUM ABSOLUTE VALUE 29 maximum absolute value of any polynomial no matter how high its degree.

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Algebraic Geometry 3: Further Study of Schemes (Translations of Mathematical Monographs Vol. 218) by Ueno K., Kato G.

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