An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski

By Oscar Zariski

Zariski presents a superior advent to this subject in algebra, including his personal insights.

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Let t~s. , s. , s. o,s. of all derivations of and this proves the proposition. If W Prop. 3: is a simple subvariety of (a) ~ W is a free r-dimensional (b) W (c) ~ W / a ~ ~ W Proof: Let D 6~W if, sn%d only if, I' " " " ~r V/k of dimension ~-module ( ~ = s, then ~fw(V/k)~ and is a free s-dimensional ~/~ -module. be uniformizing coordinates of W. , c~ I form an CT -basis of are linearly independent and the module is free because the over k(V). ~, This proves (a). (b) is obvious. Let where the ~I ti be chosen so that ~s+i are uniformizing parameters of W.

From the preceding "local analysis" it follows that F t is a non-sinEalar surface (in particular, ~ pt is a regular local ring for each point of A'), and that to the curve ~irreducible curve there will correspond on F' F ! ~B h where the B i an are the points Pl' "'" Ph which cat~espond to the tangent directions of We recall that p1 [- at P. E(~-) w~th center contain only valuations with the s~me tangential direction, ~'e recall that { x~yS Let P1 be one of the h is a set of uniformizing parameters of M.

K[y]. This shows that (Yo' '"' Yn )h ~ is integral over This shows that which shows that and b0/yiqE k~i], &O(yo,. "" ,yn )NC . , y n ) h C Assume is integral over k~i]; A i. ~ " Let Ai = k~i], Then ~yiq+hg R ~ yi q and so (Yo' ""' Yn ) ~ and let ~k(V) is integral over k[y] ~ y i q+h I is a form of fo~ 9 ~ -59degree q+h. Therefore ~ E k[Vi] which proves closed and so V is normal. Prop. 10: is normal if and only if V Proof: Assume C ~ is normal. for some Therefore h. L'q = Zq Assume say V q ~ h.

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