New PDF release: An Introduction to the Theory of Algebraic Surfaces

By Oscar Zariski

ISBN-10: 354004602X

ISBN-13: 9783540046028

Zariski offers an outstanding advent to this subject in algebra, including his personal insights.

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Extra info for An Introduction to the Theory of Algebraic Surfaces

Sample text

And if and only if 7" has no slngul=rlties. We are thus led to the following Def. s P ( z ) of Z is d e f i n e d by ~ot___e_e = (a) p(o) = l (b) p(K) = (Z 2) + l (c) p(-z)- I . Prop. z) + (z2) = 2p(z) - 2. z 2) - l. (b) p(Z) is an integer. , a cycle, then p(Z) ~ p(Z) = 7F(Z). prime Furthermore 7/" (Z) if and only if Z has no Bin~ular points. Z2)+(Z22)+2. Z 2) - z. (c) has a l r e a d y been proven0 -51(b) By virtue of If negative of a curve. that p(Z) is an integer. a curve. pY o2 t h e is a curve we have already Pence assume Tr using z = p(7') + p C - a ) = a non-singular I F I"), of the (a), I/~+KI (the 177+Zl + 1.

For we have t ~ M s+l for some integer 7EB. ~I" Let Yl = y/x~ then Yl but even ~n M I. We claim that x and Yl is not only in form a basis of ~ . and -45Let ~ - ~/~6~. lqe can write where bee M. Since M = ~(x,y), ~xn~l(X'Yl) ~ = b~ we see that ~ M = Olx. Hence 1 ~ = xn~l where ~16 ~l(X'Yl >~ ' and so we have We can write ~q = ( ~ ) x n. Since ~ M to (YI)' we have ~ / x n 6 ~I" +blXn'~ + ... + b n ~ a and Y = 0 is not tangent Therefore ~ = ~ i xn is a unit in I" Hence E/~ 6 q ( x , Y I) which shows that x and Yl generate N I.

Zvo with coefficient J and so v (Aj(~)) >-J~. F v (~1') > - ~ P" propositlon~ o Z(f) Ci = E(Yi) , for (~ > O). T%Aen vr (vj/vo ) denote Zf. Prop. re ~iYiq = ~jyjq ~i6k(V). }o ~i' and NV ~ S. S Let ~ we have are among the components of is a minimal prime ideal in S. Ci where Then for all i,j. >'-~ > 0 which proves the o- (yjq/yiq), and for a suitable choice of the hence the poles of (2) Lq. , n. , n. 8: M is be the integral v such that Then ~ = ~ v~ Rv S c Rv aud s~_uce -58v( ~o ) -> 0 for all ~o yu~ ~ R' v N ~ for some u o.

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An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski


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