An Introduction to the Theory of Algebraic Surfaces: Notes by Oscar Zariski (auth.)

By Oscar Zariski (auth.)

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Extra resources for An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58

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In its I I C R' . h hq are integral over R and R'hq To show where I' = R' , we must show that h hq I. Let ~ be in ~ q . in the quotient field of I + ~+ R' hq is contained Then = o and therefore s~w II = Since ~ ~ 0' + R 2q t + then s i. e, I. Hence we have to 9 is a finitely-generated R-module, we can write R' = Rz I + ... , st). > s we clearly have be any non-negative integer. SlZl + 9 zi Let J -56It follows that we also have Rt - RtRt. In particular, R' = r=t ~h q+J q J qh ~qJ ' IY = k + Rq + (Rrq)2 + ...

D ~ r . , ~ independent of the choice of the differential of degree r fotmizing parameter of W, Let curve on F F. [- is a prime cycle. - _ r - l ~ o = Ad~l... ~i" Thus if which is regular on then Tr~ ~ Let 00 [- . d~r_l W and if t (6J). Finally, let ((4) + r" _ (t) ~ be an irreducible t c is a cycle which does not have tO1 s(F) is. independent of OJ and t. be another differential satisfying the same conditions as We can write COl= A~ where Ae~, A4~. Choose is a set of uniformizing coordinates of Then k(r').

Let ~ ~. Hence P be the center of the which is a contradiction. Hence A~ F and F ~ i w luation any subring of is a multiplicatively ; for if f = g/h ~ ~rs, g, h ~ (I/h~)(hDg - g D h ) ~ ~ . , Dq)~ ~p , showing that ~ q Assume (2). Let P' P ~j is regular at P. q Now, let P be any irreducible (r-1)-dimensional subvariety of V such that Pg P. It is immediately seen that the integral closure of the local ring (~r(V/k) is the ring of quotients of to the ultiplicative system are regul~r at ~-. , Dq)~ ~ .

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