Download e-book for iPad: An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski

By Oscar Zariski

ISBN-10: 354004602X

ISBN-13: 9783540046028

Zariski offers an effective creation to this subject in algebra, including his personal insights.

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Additional info for An Introduction to the Theory of Algebraic Surfaces

Sample text

Xr§ Let : p. , Xr§ A and B and wh(x) awh = are of smaller degree ~ -61- Xr+ 1 then the equation of integral dependence of Xr+ I Since and hA = gB, h this relation must hold for the are not associates, we have ~ = c(x) is in R. Let V Therefore be a variety, let the integral closure of R k(V), RI 9 %hen Then K = Ko(Yo). R'~ , Let r Let the v R ~t ~ 9 : ~ g and . be its coordinate ring and let R t be K. If we let Ko denote be a minimal prime homogeneous ideal iu v' v' of K. Since has transcendence the variety of the prime ideal dimension as ~ '.

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 .

This shows that if Thus P on F, ~. Let { x,y} and let t = 0 Again we can let 03 = dxdy, be a set be a local equation and so we have as before (t)p = [" and ( ~ ) p = O. Hence - (Tr tr Tr ((~s) + [" . (t))p- (TrFt ~O)p = fbe the set of prime divisors Y of k ( F ) which dxdy)p. Let B are centered at P. Let ~ ~ ~r t dxdy I r (~)p ,. __ We can consider exist uniqae elements lcok on M/M 2 M/M 2 c,d as a vector space over such that ~ k. For ~ ~ M, - (cx + dy)~I~ 2. as the two~limensicnal set of linear forms there Thus we can cX + dye The elements of the corresponding one-dimensional projective space are definable by equations of the form cX + dY = 0.

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An Introduction to the Theory of Algebraic Surfaces by Oscar Zariski

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