POLYNOMIAL RING S AN D AFFIN E SPACE S 7 o{g/h \g, hek[U], h(q) * 0 } = {g/h \gt h e k[W] , hip) * 0 } where q i s a point correspondin g t o p. Next w e observe projectiv e varieties . A homogeneous coordinat e rin g of «-dimensiona l projective spac e Pn i s a polynomial rin g H = A:[X 0 , . . . , Xn] i n « + 1 variables ove r th e ground fiel d k. I n thi s case, we employ homogeneou s coordinate s an d therefore : (1) Element s o f H are not function s o n Pn, (2 ) i f f(X) i s a homogeneous for m i n X0, . . . , X n an d i f p = (p 0 , . . . , pn) i s a point o f Pw, the n i t i s well define d whethe r o r not f(p) = 0 , and therefor e (3 ) i f / i s a homogeneous idea l o f H (i.e., / i s generated b y homogeneous forms ) the n th e se t V(J) of zero points of J i n Pn i s well defined , an d (4) if f(X) an d g(X) ar e homogeneous form s o f th e sam e degree , then f(X)lg(X) i s a well define d function o n Pn - V(g(X))\ suc h a function f/g i s called a rational function o n Pw.H A = Pn - V(X ( ) = {(p0, . . . , pn) GP n \p t = 1 } is naturally regarde d a s an ^-dimensional affin e space with coordinate rin g Rt = k[XjX v . . . , X^JX^ X i+1 /Xt, . . . , A^/JTJ, / " = WQ U UH/ fl unde r th e birationa l correspondenc e betwee n H A an d H A define d b y tha t R i and R- have th e sam e fiel d o f fractions , p ( G HA ) corresponds t o (7 ( E HA ) if an d onl y if p = q in Pn. Thu s Pn i s the union o f n + 1 affine space s and gluin g them i s made via natu- ral birational mappings . I f P is a homogeneous prim e idea l o f Hy then U = F(P ) is a projec- tive k-variety in P" an d £/ " is the unio n o f a t mos t n + 1 affine ^-varietie s whic h ar e naturall y birational wit h on e another the commo n functio n fiel d i s called th e function field o f U and the rin g H/P is called th e homogeneous coordinate ring of U over k an d H/P wil l be denote d byHk[V\. If W is an affin e o r a projective ^-variet y an d if a n injectio n o f k(W) int o k(U) i s given, then w e have a rational mapping (rational correspondence) defined naturall y vi a affin e pieces . Its regularity at a point, birationality t biregularity at a point ar e define d similarl y a s in th e affine case . Now, one for m o f th e completeness o f U is stated a s follows : THEOREM 2.1 . / / U and W are as above, then for any point q of W, there is a point p in U which corresponds to q and therefore if W is also projective and if Z is a closed subset of W then the set Z' of points of U which corresponds to some points of Z is a closed set. Since th e discussio n o f this kind i s not ou r main subject , w e omit th e details . Wha t we wish to emphasiz e her e i s that th e importanc e o f th e completenes s lie s in th e fac t tha t i n order t o investigat e propertie s o f birationall y equivalen t varietie s w e often hav e a lot o f help given by th e propertie s o f f/vi a birationa l correspondence . These examples , Bezou t theore m an d completenes s sho w good feature s o f projectiv e varieties. Bu t ther e ar e many example s by whic h we feel tha t projectiv e varietie s ar e muc h more complicate d tha n affin e varieties . On e typica l exampl e i s the produc t o f spaces . Th e product o f affin e space s Sm an d S n ove r th e fiel d K i s nothing but (m + n)-dimensional affine spac e Sm+n\ th e pai r o f p = (plf . . . , pn) an d q = (qv . . . , q n ) i s the poin t iPv - Pm Qv O - B u t i n t h e projectiv e case , if p = (p 0 , . . . , pm)GPm an d q = (q0, . . . , q n ) G Pn, simila r arrangemen t o f th e coordinate s make s (p0, . . . , p m , 4o - - 4m) G P m + w + 1 whic h i s not uniquel y determine d b y p an d q (because , fo r
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