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Additional resources for An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58
D ~ r )" Thus there are only a finite number of prime divisoria! cycles are not uniformizing coordinates of ~ . , r. , r a~ Denote the right-h~d side of (*) by s(t--). si o sd'-i). Hence each Ai Z o "• , 0 CJ . Let C~ ~ I--1, . . , ~tl--i" (*) l--h~ , We have sho~n ~ t (A•247 k, -~0, and . Trace of a differential. ), ... varies in a finite-dimensional vector space over h e n c e so does w at A i, namely vp(A i)- a ~ let ~ I' "" 9r be a subvariety of V; ~ = k(V). Let ~ at W; then ~ Y and let ~ = ~w(V/k), ~ = ~w(V/k), be the set of all derivations which are regular is an ~-module.
Is integral over of degree N. , OJ~ Rq, Let Let Then for large ~ we have ~ y i ~'q a R ~. For ( ~o ) = (uJ/yoq) = D - qC o. the theorem. and th~s ~ i Y i U i c R I for each i. all mono1~ials in Yo' "" "' Yn and so ~ c ~o m ~ Lq is complete q~ and R ~ be as before, and let ~ be the conductor of R' is a homogeneous ideal and hence is graded. Therefore ~e can = J[o + ~ I + "'" + ~ h + "'" where ~o # @ if ~ is the unit ideal. 9: V is normal if and only if ~ contains a power of the irrelevant prime ideal (Yo, " "~, Yn )' Proof: Assume V is normal.
T we see that ~ ~ C t. ~i = v/~n ~. We shall show that ~ = g / ~ s bl ~ ~ Th~ef=e ~' ~ = ~'t where ~ ' e ~ . ~l ~ ~I' s ~e~. is a u~t in ~, ~ ~#-s ~l where and ~= :~l_ni~ltl. Hence ~ = ( ~ l / ~ l ) t l ~ ~itl , and so ~I" This shows that ~ = 0 is a local equation of ~- in ~I" What we have described above is the effect, in a given direction at ~, of a so-called "locally quadratic transformation" center P. A full (global) description of Let Yo' " ' " Yn general point is the point A of F/k, At k.