8 MASAYOSHI NAGAT A instance, q = (tq 0 , . . . , tq n ) wit h a n arbitrar y t =£ 0). Therefor e (p 0 , . . . , pm, q 0 , . . . , qn) doe s not giv e the pai r o f p, q. On e goo d way t o dea l with th e pai r i s to assig n a sequence o f product s o f pt an d q j9 sa y (p 0 q0, p x q0, . . . , pmq0, p 0 qv . . . , P m qv PoQi* - - - Pm^n^- I * *s eas y t o se e t n a t ^ t m s assignmen t w e n a v e a g°° d biregula r corre - spondence betwee n P m x P n an d a certain projectiv e fc-variety in pmn + m +n. For th e purpos e o f ou r later discussion , we add som e discussio n o n birationa l map - pings. Assum e tha t k i s a field a s before an d tha t U is a projective fc-variety in Pn. I f a finite numbe r o f element s z0, . . . , zm o f k(U) i s given, then w e have a Avariety W and a rational mappin g o f U toI V as follows. Tak e a transcendental elemen t t ove r £(£/) an d con - sider th e natura l homomorphis m o f a polynomial rin g k[Y0, . . . , Y m ] ont o fc|z0, . . . , fz m ]. Th e kerne l Q is a homogeneous prim e idea l an d w e haveI V = F ( 0 . W is the unio n o f affin e ^-varietie s wit h coordinat e ring s Af = k[z 0 /z(, . . . , zm/zi] whic h ar e subrings o f k(lf), henc e w e have a natural rationa l mappin g o f U to W. W is called th e pro - jective fc-locus of (z 0 , . . . , zm). Th e sam e is applied t o a sequence o f element s z0, . . . , z m such that al l zjzj ar e in k{U). Now, letting fc[jt0, . . . , xn] b e th e homogeneou s coordinat e ring Hk[U] (an d (/i s the projectiv e A:-locu s of (x 0 , . . . , *„)), we consider th e pai r o f (JC 0 , . . . , xn) an d (z0, . . . , zm), i.e. , (x 0 z0, . . . , x„z0, x 0 z p . . . , x„zw) i n P w " + w + ". Le t T be th e pro- jective fc-locus of (x 0 z0, . . . , ^„^m). A s is easily seen , the birationa l mappin g o f T t o V is regular everywhere the inverse , i.e., th e rationa l mappin g o f Uto T, is called the dilatation of Uby(z09. . . , z m ) . If W is birational t o U and i f W — U is regular, then T is biregular t o W. Therefor e any projectiv e variet y W such that W —• U is birational an d regular , is biregular t o a projec- tive variety obtained by a dilatation. This means that general dilatation is very difficult t o deal with. But , in som e specia l cases , dilation behave s nicely. On e see s easily THEOREM 2.2 . Ifz 0 , . . . , zm are taken from H k [U] so that they are homogeneous elements of the same degree, then the rational mapping of U to T is regular outside of V&ztHk[U]). The simples t cas e is therefore th e cas e where Xz i Hk[U] = I(p) wit h a /^-rational poin t p = (p 0 , . . . , pn) (i . e., Pj/pj £ k fo r al l possible pair s (i, /)). The n U —• T is regular ex- cept at p. This dilatation is called the quadratic dilatation with center p. I f one checks carefully , one see s easily tha t th e se t E o f point s o n T which correspond t o p i s a subset o f T whic h is biregular t o th e bas e variety o f th e tangen t con e o f U at p\ i f p i s a simple point the n E i s biregular t o pr (r = di m U - 1 ) and every poin t o f E i s a simple poin t o f T. Quadratic dilatations play a very important role in the theory of surfaces. On e reason is the validity o f th e followin g THEOREM 2.3 . / / U and W are birationally equivalent nonsingular projective surfaces and if the birational mapping W — U is regular, then W is biregular to a surface which is obtained by certain successive quadratic dilatations. In th e highe r dimensiona l case , the situatio n i s much mor e difficult . Fo r instance , it is usual tha t th e se t o f points a t whic h a given dilatatio n i s not biregula r ha s some positiv e

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