POLYNOMIAL RING S AN D AFFINE SPACE S 9 dimension. Therefor e we are obliged to observe dilatation other than quadratic dilatation. A n immediate generalizatio n of a quadratic dilatation is a dilatation define d by (z0,. . . , z ) such that Sz,.//^!/ ] i s a prime ideal,o r (a prime ideal ) O (a primary idea l belonging t o the ideal generated b y all positive degre e elements ) (under the notation of Theorem 2.2). Such a dilata- tion i s called the monoidal dilatatio n with center C= V(Ez t Hk[U]). I n thi s case, if C is a nonsingular ^-variet y an d if every poin t o f C is a simple point of U, the n the dilatation appears to be understandable. But , in the general case , phenomena alon g singular subset s o f C are very complicated . O n the other hand , if Xz i Hk[U] i s the intersection o f certain prim e ideals, then som e complication happen s a t the intersections o f the components of VQlZiH^U]) a s one sees easily b y the following example : Consider P3 wit h homogeneous coordinat e syste m (X 0 , X v X 2 X 3 ). Tak e line s Lt = V(XjH + XmH) {H = k[X 0 , . . . , X3] ft / , m} = {1, 2, 3}). The n I(LX UL2 UL 3 ) = (XXX2, X 2 X3, X 3 Xt). The n we define the dilatation/?3— W defined by (XXX2, X 2 X3, X3XX). W has four exceptiona l surfaces , on e of which correspond s t o the origin (1, 0, 0, 0). Furthermore, (this dilatation cannot be factored int o a product of successive monoidal dilatations. In connectio n wit h thi s fact , th e writer like s to note tha t th e following i s an important question o n birational mappings : Question 2.4. Assum e tha t U and W are birationally equivalen t nonsingula r projectiv e varieties. Doe s there exis t a projective variet y T such tha t bot h o f the birational correspon - dences U — T and W — T are factored int o products of successive monoidal dilatations with nonsingular centers ? In closin g this section, we give a few remarks on correspondence o f subvarieties unde r a birational mapping . Firs t o f all, we should no t restrict ou r observation onl y t o set theoret - ical aspects . Fo r instance, a divisor is a linear combinatio n o f divisorial subvarieties (i.e., subvarieties of one dimension less ) under certai n restriction s (no restriction i n the nonsingu- lar case), and by virtue o f the notion, we can discuss the Riemann-Roch theorem , fo r instance. Therefore, let us observe ideals instead o f subvarieties in the projective case , it suffice s t o observe homogeneous ideal s of the homogeneous coordinat e ring , in the afflne case , just ideals of the affine coordinat e ring , and if we want t o observe mor e genera l case s (abstract varieties or schemes) w e observe sheave s of ideals. I f an ideal is locally principa l everywher e it define s a positive Cartie r divisor . But , in the general case, the geometric featur e o f an ideal is not very clear yet and must b e one object t o be studied in the future . Anyway, if W and U are varieties and if a regular rationa l mappin g o f W to U is given, then a n ideal J on U generates a n ideal f on W (in the projective case , irrelevant primar y components may be disregarded). On e disadvantage i s that tw o mutually distinc t ideal s on U may generate th e same ideal on W. Bu t this does not happen i n certain goo d cases , in- cluding the case of ideals for divisor s on U if U is projective nonsingula r an d if W is projec- tive. Anyhow , we have a mapping of the set of ideals on U to the one on W associated wit h the regula r rationa l mappin g W — U. B y this mapping, we define total transform. Fo r instance, if U and W are projective an d if U is nonsingular, the n fo r a divisor D on U, D is expressed a s Dx - D 2 wit h positive divisor s Dt whic h correspond t o locally principa l ideal s Jv J 2 o n U and they defin e Cartie r divisor s D[, D 2 o n W by the mapping give n above. Then th e total transform of D is defined t o be D\ - D 2 .

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