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An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski

By Oscar Zariski

Zariski presents a high-quality creation to this subject in algebra, including his personal insights.

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An Introduction to the Theory of Algebraic Surfaces

Zariski presents an effective creation to this subject in algebra, including his personal insights.

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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 + ...

6: Any two non-singular models V,V' field ~ (= k(V) = k(V')) of a given function have the same geometric genus. d. Z/k = r. We define the gepmetric genus of ~/k, p (Z/k), to be the minimumvalue of p g(V) g ~here V is any proJective model of ~/k. 7: If V/k is a normal variety, then the differentials of a given degree q which are regular on V form a finite- dimensional vector space over k. Proof: We ~all assume q = I since the proof is the same for any q. Fix a separating transcendence basis { ~i' "'" ~r ~ of k(V)/k, and let &2 be any differential of degree I which is regular on V.

First kind with respect to V rise to a min~al homogeneous prime ideal in R'. gives Thus we have a (l-l) correspondence between prime divisors of the first kind with respect to V and minimal homogeneous prime ideals in R t . Again let let ~ ~ I be a minimal homogeneous prime ideal in R t, be the corresponding prime divisor. and Since ~ v is homogeneous, co we can write ~ ' =m=l ~* ~B tin where, of course, ~ m the associated linear system, and let LS(~'m). , LS(~Vm) Since L' m m LS(R') = L' LIA m" It is easily seen that is the set of ~--residues of is complete, we know by Theorem %o7 that q.

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