By Oscar Zariski
Zariski presents a pretty good advent to this subject in algebra, including his personal insights.
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Additional info for An Introduction to the Theory of Algebraic Surfaces
The proof of the following proposition is obvious and we omit it. Prop. 11: Let V be a projective variety, and let i ~''" ~ ~t' (a) (b) (V)= ~ o ~ where T o is either (I) or is primary for ~ i ~ is a homogeneous primary ideal Let V a be an affine representative of V (Yo' "'" Yn )' (~i = ~ i and let )" P 6 V a. Then -6o- (1) t (Va) = ~ ~i, dh i=l t ~ ~, dh (2) /~p(v) o r] ~ i=l (3) dh = (~) if and o~ly iZ V( 9i ) i~ oon~i=od in V - V a . ~s ~4ith the conductcr of V. ~ It follows that Cor. ~mal at P we associate the subvariety if and only if is normal at if and only if P V V(~ ) is normal.
Let L of the regular differentials on V, L -~0}* IK# , where car as a projective space. Since fact, be the vector space (over k) of degree car), % dim (K~ / o. < | dim L -- I + dim ~KJ . We denote r. genus of the variety V. p g(V) linearly independent differentials of degree on V. Clearly, If V pg(V) We have a mapping dim L < =: IKI in p g(V), and call Hence V r carries exactly which are regular is al~ays => O. is biregularly equivalent to V', the pg(V) = pg(V'). This is not necessarily true for birationally equivalent varieties.
Xn) rank n-r on W, hence we can assume ~(fl' "''' fn-r ) # 0 on W. , X r ~ is easily seen to be a set of uniformizing coordinates for W. Prop. ~r) ~ ~"W" a( ) Proof: Assume the ~i ~(~) 3(El and (b)implies = I, are uniformizing coordinates. ~(~) ~(~) Then ~. Hence ac ) is a unit in ~ . , r, AijE K. , z m] contained in field). k(~) = k(g) We call ~ on V (hence R 9 d i m W = r-l). such that W V a . , Zm) a (~) AijE~. c 8 ~ since R = k~Va], be t h e i n t e g r a l c l o s u r e of be the locus of variety, and Di ~ K/k Va holds.
An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski