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Additional resources for An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58
R. d~r = . Then i ~(~I~ "'" ~r)id d~r. '~(~i' "'" ~r ) NI"" are uniformizing coordinates, we have v (d~l... d ~ r ) = v ( B ( ~ ) J = coefficient of ~ in the divisor r d~r). Since the ~ i are not uniformizing coordinates of is infinite at ~ or ~ is a component of cycles r . Let eihher some ~i (d~ I ... 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 R be the co- ordinate ring of V, and let Rt quotient field. Ps Let ~ (V) and let W = ~p(V/k) = ~r(V/k), Let H(Y) = CoYo + ... + C n Y n = 0. -~ ~ Let V Let Let . We define ~ d h i S an ideal in mapping q, (Yo''"~Yn)h ~ ~ Zet R = k[y], and let 07 be any homogeneous ideal in R. Zet V a = V - V ~ H q~ ~p(V) be the integral closure of R denote the conductor of R' be the locus of and let ~' P over in its in R. K. Let be the integral clo2ure of denote the conductor of ~ ' in ~ . The proof of the following proposition is obvious and we omit it.
S+C[ r-s s+i = ti, ~ ~ti, i=l r-s on W. , r - s. 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 I' " " " ~r D 6~W if, sn%d only if, 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).