By Oscar Zariski (auth.)
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Extra info for An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58
Dq+ I)). exists a differential P ~q-1 such that V/k. e. 2: Oq ~ = ~p(V/~), ~ = a~p(V/k) and be uniformizing parameters of m~ = Y ~ ti). i=l transcendence basis of P on V Then ~ tl, ... trl is a separating K/k, and a. , r. Proof: To prove this theorem we shall use the following well-known result (which we state without proof): (3) If an extension over k , k(x) of a field k it is separably algebraic over In view of (3) we must show that if tr) (hence D is trivial on is the quotient field of on ~ has no non-trivial derivation .
12~2: The varieties V and Vt birational transformation rational transformation Proof: Since the ui/u j ~ K o ui for all and all from considering the elements combinations of the u j) k(V} = k(V' ). V Let A I9 are birationally equivalent, the V: -~ V V -* V I is regular and the bihas no fundamental varieties. , y o q ' ~ n that and Vi = V - (V~Hi) k ( U ) c K ~ ~ k(V). , n, Vt It is clear and therefore are birationally equivalent. , n. , n. H lI. ,Um/yqS~ k~oS. u~ + al(~)u~'l + ... e. , n. (a) We have Let aj(y) + al(Y> yq q, where ui i~ in ai(Y)E R.
AD. is the - fi ) = D ~ g i then D~t. _ O (mod A) in ~ . This implies J Hence D ' ~ a / ~ ~ ( A + ~ , i) = ~A. Therefore i=l D I~ ~ ~ . A D is for all Then A i. in D ' ~ l"+ l C ~ : D'k[t] C A k [ t ] C A ~ D~E of So we may Since /~ ~qi = (0). i=l S / S t i. There exists an element D' = AD, D there exists a polynomial quotient ring of a finite integral domain, there exists an element , K , we need only consider the effect of then, for any kit] k, . i -26- Def. 3: A derivation DRv C R D where v is regular at a given valuation R is the valuation ring of V Def.
An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58 by Oscar Zariski (auth.)