By S. Iitaka

The purpose of this booklet is to introduce the reader to the geometric conception of algebraic types, specifically to the birational geometry of algebraic varieties.This quantity grew out of the author's booklet in jap released in three volumes via Iwanami, Tokyo, in 1977. whereas scripting this English model, the writer has attempted to arrange and rewrite the unique fabric in order that even rookies can learn it simply with out concerning different books, comparable to textbooks on commutative algebra. The reader is just anticipated to understand the definition of Noetherin earrings and the assertion of the Hilbert foundation theorem.

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Then we get a "universal" I I c Rj^^) ® R^^^[s,t]/(s2-f(t)) defined by the same formula {*) with uf-*"^ ,vi^^ € RJ-*-^ , v{^\v[^^e being variables. 36 becomes a (C-vector space of fixed dimension N. °'^ basis of A, -module. M ID f. > Tc. e. i < N-g+l" defines r 1' 2 . in the open set h ^ 7^ 0 of Zxz 4'This follows from the Proposition. If R is the affine ring of an affine variety, S a finitely generated R-module, and dim^ S 0j^ R/tTV is constant as m varies among the maximal ideals of R, then S is a locally free R-module.

0 ) . v. V. )+U, Vp = 0, while for £ = 0 we get - I k=l t (P)^"^t^V, ) . 1. P ft Supp -* I ^', i=l ^ For the proof, we also assume and that neither P nor any P. is a branch point. ^ The result will follow by continuity for all P and J]P. Let I P. correspond to (U,V,W) as usual and note that as no P. is a i=l ^ ^ branch point, U,V have no common zeroes. )+ i(P) so q has poles at J P. and at P. •• At infinity. 47 So the equation or q(s,t) - j U(t)(t-t(P)) = |[U(P) Cs+V(t))+U(t) (s(P)-V(P))] has solutions!

1 = t(P^) 2 = t(P n ) has codimension 2 in (C^) , so s(P-,»«,P ) has no poles at all in (C^) . s(P^,--,P^) . Thus the coefficients V. ) and s (P^ , • • • ,P, ) , hence are functions in that (C^) > V(a ) is a morphism. 25 A consequence is that the set V(a ) is irreducible since (c^) maps onto V(a ) and (C^) is irreducible. ,•••,£,) is the prime ideal of all polynomials zero on V. (Proof omitted). We want to emphasize at this point the rather unorthodox use that we are making of the polynomials a) U,V,W: we have a bijection /divisors D on C \ *of a certain type/ ^ ^ /three polynomials \ VU(t) ,V(t) ,W(t) of a certain type/ Thus b) these divisors D become the points of a variety for which the coefficients of U,V,W are coordinates.