# An Introduction to the Theory of Algebraic Surfaces: Notes by Oscar Zariski (auth.) By Oscar Zariski (auth.)

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Conditions are satisfied. , . . , ~r ) r. contains a set of uniformizing V/k. on Furthermore, (b) is equivalent to (b') Let ~ q of A~q = NWw(V/k). Then a~q f] k[ ~ ] contains a basis . Proof: Assume (a) and (b). Fix uniformizing coordinates ~ I' "'" ~ r of W on V/k. By virtue of Prop. l are in ~ , say A l ~ , r >2. A such that ~i~ r Let D = F Aj Since the ~ j are uniformizing coordinates, we have D is regular on W. D(k[~])~4h~. and so D has a trace on ~ = Try. ~ it fellows that c /~4. Therefore D~iE,Y~ implies This shows that the Dg~W ~ ~i = O.

And V' A'/k L = k(A) = k(A t). Let A' We say exists a wluation R' = R ' ~ and Hence Kq DI and q are homogeneous. k for each q. , urn) as the coordinates of It is clear that they are strictly homogeneous co- Let respectively. Ko[Yo]. is a finitely-generated R-module, so where we can assume the R'q = ku o + ... ieties if there such that B is the center of v on A and -53B~ is the center of A ~ B v on At. We say the birational transformation has no fundamental varieties if to any irreducible sub-~ariety of there corresponds only a finite number of irreducible subvarieties of Prop.

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 .

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