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A NON-DIVISORIAL VARIETY by Robert Joseph  Fraga  B.A., Pomona C o l l e g e , 1961 M.A., U n i v e r s i t y  o f B r i t i s h Columbia, 1963  A t h e s i s submitted i n p a r t i a l f u l f i l m e n t o f the requirements f o r the degree o f DOCTOR OF PHILOSOPHY i n the Department of Mathematics We accept t h i s t h e s i s as conforming to the required  THE  standard  UNIVERSITY OF BRITISH COLUMBIA June, 1965  In p r e s e n t i n g the  r e q u i r e m e n t s f o r an  British  Columbia,  available for mission  representatives,,  cation  of  w i t h o u t my  this  study.  by  the  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Columbia  of  University  of  s h a l l make i t  thesis  Head o f my  permission.  fulfilment  I f u r t h e r agree that  this  for financial  the  Library  It i s understood  thesis  written  the  copying of  granted  in p a r t i a l  advanced degree at  r e f e r e n c e and  be  thesis  I agree that  for extensive  p u r p o s e s may his  this  copying or  shall  not  per-  scholarly  Department o r  that  gain  for  freely  be  by publi-  allowed  i  Advisors  Dr. Mario B o r e l l i  ABSTRACT  When d i v i s o r i a l v a r i e t i e s were f i r s t introduced,  the question  immediately arose whether there are any v a r i e t i e s which are not divisorial.  T h i s work answers the question i n the a f f i r m a t i v e .  We prove here that the n o n - p r o j e c t i v e  v a r i e t y M defined by  Nagata i n Memoirs o f the College o f Science, U n i v e r s i t y of Kyoto, S e r i e s A, V o l . XXX, Mathematics No. 3, 1957, pp. 231235 i s , i n f a c t , n o n - d i v i s o r i a l .  The work i s organized  as f o l l o w s *  We f i r s t  discuss  briefly  the concepts r e l a t i n g to the notion of d i v i s o r i a l v a r i e t y .  Next  there i s a d e s c r i p t i o n o f Nagata's v a r i e t y i n which we i n c l u d e the proofs of statements which we s h a l l need f o r the subsequent theorems.  The p r e l i m i n a r y r e s u l t s are of two types:  prove s e v e r a l lemmas concerning o f p o i n t s on the v a r i e t y M.  First  we  the dominance of l o c a l r i n g s  5econd we prove that d i v i s o r s whose  v a r i e t i e s c o n t a i n the vertex o f the a f f i n e cone V used i n Nagata's example must i n t e r s e c t the l i n e at i n f i n i t y o f the cone at a p o i n t ( s ) whose l o c a l r i n g dominates the l o c a l r i n g of t h e vertex of the cone  under the transformation cr defined by  Nsgata.  This r e s u l t i n d i c a t e s s t r o n g l y that the v a r i e t y M • VU V°~ i s  ii  not  divisorial.  For the proof t h a t t h i s i s , i n f a c t , the case,  we prove i n d e t a i l a s t r i c t l y a l g e b r a i c r e s u l t to the e f f e c t that the (prime) i d e a l s a s s o c i a t e d with the i r r e d u c i b l e  com-  ponents o f a d i v i s o r which do not c o n t a i n the vertex P o f the cone V are p r i n c i p a l . contradiction  With t h i s r e s u l t , we f i n a l l y  that M i s not d i v i s o r i a l .  show by  iii  ACKNOWLEDGEMENTS  I would l i k e t o take t h i s o p p o r t u n i t y to express my thanks to Dr. Mario B o r e l l i , who was my t h e s i s a d v i s o r and whose encouragement and d i r e c t i o n a l i n s i g h t f a c i l i t a t e d tion of t h i s thesis.  I would l i k e  the produc-  to thank Dr. W i l l i a m  who k i n d l y consented to read t h i s paper.  Hoyt  Also my thanks are  due to the U n i v e r s i t y o f B r i t i s h Columbia, the N a t i o n a l Research C o u n c i l o f Canada, and Indiana U n i v e r s i t y without whose  generous  f i n a n c i a l support t h i s work would not have been p o s s i b l e *  iv  TABLE OF CONTENTS  SECTION  PAGE  Introduction  1  Natation  2  and Terminology  Nagata's C o n s t r u c t i o n  7  P r e l i m i n a r y Results  12  Main Theorem  27  Conclusion  30  Bibliography  31  1 1  INTRODUCTION  In 1963, B o r e l l i  defined  the concept o f d i v i s o r i a l  proved some of i t s p r o p e r t i e s  i n L"5] .  v a r i e t y and  The question arose  whether there were v a r i e t i e s which were not d i v i s o r i a l . purpose o f t h i s t h e s i s i s to e x h i b i t an a l g e b r a i c i s not d i v i s o r i a l , Since B o r e l l i  v a r i e t y which  thus answering the question i n the a f f i r m a t i v e .  proved that a l l non-singular and a l l q u a s i - p r o j e c -  t i v e v a r i e t i e s were d i v i s o r i a l , for  The  i t was c l e a r that  the search  a n o n - d i v i s o r i a l v a r i e t y had to be l i m i t e d to those v a r i e t i e s  which were s i n g u l a r and n o n - p r o j e c t i v e . structed  in  Such a v a r i e t y i e con-  by M. Nagata, who proves that i t i s a normal  v a r i e t y which cannot be imbedded i n p r o j e c t i v e prove that  t h i s v a r i e t y i s not d i v i s o r i a l .  space.  We  shall  2  2  NOTATION AND  We  s h a l l consider  closed  field.  TERMINOLOGY  only v a r i e t i e s defined  We  give f i r s t the  known r e s u l t s without  We  introduce  the  over an a l g e b r a i c a l l y  f o l l o w i n g d e f i n i t i o n s and  proof.  f o l l o w i n g "double" n o t a t i o n *  i r r e d u c i b l e v a r i e t y , D a d i v i s o r of X,  Let X be  «m  Y c U c X , where U i s an  open, n e c e s s a r i l y i r r d u c i b l e subset of X,  We  denote by  the d i v i s o r on U c o n s i s t i n g of those prime component?* of  (D)y  as they have i n D.  p o s s i b l e as w e l l as Y • U 4 X.  y  D,  r e s t r i c t e d on U, which i n t e r s e c t Y non-emptily, taken with 8ante m u l t i p l i c i t i e s  well  the  Note t h a t U • X i s  A d i v i s o r D of an i r r e d u c i b l e  a l g e b r a i c v a r i e t y X i s l o c a l l y p r i n c i p a l ( C a r t i e r ) i f there exist-;; an open c o v e r i n g f^  , i £ Ij  that D  v  «  X , A  , of the  (f.) X  v  of X,  ^  « jlL  , i£-l}  , and  a subset,  f i e l d E of r a t i o n a l f u n c t i o n s of X such  f o r a l l x £ U.  X , A  and  i£l.  Here ( f . ) denotes  1  the d i v i s o r of the f u n c t i o n f ^ on X.  X  We  remark that i f a v a r i e t y  X c o n s i s t s e n t i r e l y of p o i n t s x a l l of whose l o c a l  rings X , A  are UFD  (factorial),  then a l l d i v i s o r s D of X are l o c a l l y p r i n -  cipal. For the  f o l l o w i n g remarks, we  drop the c o n d i t i o n t h a t X  be  3  irreducible. denote of  the sheaf of l o c a l r i n g s o f X and  ft^.  called  An element  the sheaf o f u n i t s  of the f i r s t cohomology group H*(X,  T (U^j, ^J)  the set of s e c t i o n s of  is  f £ H ( X , ^ ' ° ) and l e t 1  U  - {U  A  , It  IJ  over u\j« u \ 0  represents f.  We  s h a l l say t h a t  CU,b)  »^x*  *"^ ij U  represents f .  sheaf F i s s a i d to be a l g e b r a i c over X i f i t i s a shaaf o f  O'^ modules.  I f F i s an a l g e b r a i c sheaf over X, then there e x i s t  an a l g e b r a i c sheaf K and l o c a l isomorphisms u^t such t h a t f o r every x t l l ^  K|IL  v F 'u.  and a €. F^ ,  ( u u " ) ( a ) - b(i,J)(x)- e 1  i  The  U^,  be an indexed opsn  c o v e r i n g of X f o r which the 1-coeycle bt ( i , j )  The  Let  a l i n e c l a s s o f X.  Denote by Let  Hence l e t X be an a r b i t r a r y v a r i e t y .  j  sheaf K i s uniquely d e f i n e d up to  depends only upon F and f .  ^^-isomorphisms,  and i t  The l o c a l isomorphisms u^ depend  on the r e p r e s e n t a t i o n C U , b ) .  The sheaf K i s denoted  by f ( F ) .  Since F and f ( F ) are l o c a l l y isomorphic. F i s coherent i f , and only i f , f ( F ) i a coherent.  For a more d e t a i l e d d i s c u s s i o n of  these matters, the reader may  consult  4  Assume now that X i s an irreducible variety. be an open covering of X, and l e t m ideal of  0"^ ^ for every x€ X.  Let Zi • [  , i£  denote the unique maximal  Then the subset ( f ^ . i € l j  of  rational functions o f X i a called a coherent *H-system of l o c a l generators  ift  (1)  f  (2)  f. £ (y  A  » 0 for every y  - n  for ©very x €. U, . and a l l  f,  If  «  r f  i j€X. t  and  J  (3)  i€.I,  j ' h f  for a l l i , j , h £ l . f  h  f i s the l i n e class represented by J  i , j £ l / , we say f.  that  f  ^ i s a system of l o c a l generators of f•  Suppose f,  generators.  a given l i n e c l a s s ,  h a s s u c h a s y s t e m of  local  Then we can associate an equivalence c l a n  Cartier divisors of X with f. that of l i n e a r equivalence*  t h e equivalence r e l a t i o n If  we denote by [fj  of being  the d i v i s o r  d a e a with respect to linear equivalence of X associated w i t h  •f» is  f — - H f l  a group homomorphism from H (X,<^ ) into the additive group  5  of the d i v i s o r c l a s s e s under l i n e a r equivalence of X whose d i v i s o r s are l o c a l l y p r i n c i p a l .  This homomorphism i e  iso-  morphism i f X i s normal.  f(&^)  A l i n e c l a s s f o f X i e s a i d to be r e g u l a r i f the sheaf has a non-zero  s e c t i o n over X.  an isomorphism  between •  P  I f f i s r e g u l a r , there e x i s t s  (X, f ( E )  )  jjf|  and  notes the non-negative d i v i s o r s i n \f\ . such an isomorphism  always e x i s t s ;  The reader may  consult  jjfjj  de-  remark here that  the requirement that f be  r e g u l a r guarantees that the isomorphism sets.  We  where  i s not between empty  \A~\ , § 5 , f o r the p r o o f s o f these  statements.  The a t a l k o f fifty)  over any p o i n t x C X  has a unique maximal  submodule, which i s denoted by n^ , corresponding to the unique maximal i d e a l n»  of  Given a s e c t i o n  mC  X, A  X  we  .  v  A  define  \ »{*€X(s(K)4n } x  We  remark here that X  i s an open set i n X. 8  Let dj (X) denote the c o l l e c t i o n of open sets o f X. B  k  - (UC^(X) \ U • X  B  , e c H x , f ( ^ ) ), x  Then we  fCH^X^jjJ  define  6  Definition  An  algebraic  constitutes  The  significance  the  preceding remarks*  variety  a base f o r the  of X,  X,  whoee v a r i e t y  a closed  subset Y of X such,  t h a t  d i v i s o r of X which i s l o c a l l y  c o n t a i n s Y but  i n e f f e c t , i s determined by  are l o c a l l y p r i n c i p a l *  of  I f X i s an i r r e d u c i b l e d i v i s o r i a l v a r i e t y ,  there e x i s t s a p o s i t i v e  p r i n c i p a l and  topology of  of t h i s d e f i n i t i o n becomes c l e a r i n view  then given a p o i n t x £ X and x^Y,  X i s called d i v i s o r i a l i f  not -x.  the p e i i t i v i  The  topology  d i v i s o r s which  7  3  NAGATA*S CONSTRUCTION  3  3  Let V be the cone defined by X  Y  3 • Z  over the f i e l d o f  V may be regarded as the r e p r e s e n t a t i v e  r a t i o n a l numbers R.  cone o f the p r o j e c t i v e curve V* with where a and b are transcendental  generic p o i n t D* « i a , b l ) 3 3 p  numbers such that a  + b  *> I .  Let L e tKk*beR C s ^ f e ) • an a D l g ebe b r athe i c a lgenerator l y c l o s e d o ffi eVl dwhich c o n t agoea i n i n gthrough a and D*. k, 4*»». Consider the image o f V under the by the f o l l o w i n g  affine transformation defined  equations!  mx  X  + az  Y « y + bz  m Z  Z  We denote t h i s image aa V without f e a r o f c o n f u s i o n .  Vis  thus d e f i n e d by x  3  + y  Let & «> k [ x , y , z j  3  . d  + 3(ax  2  • by )* * 3(» x • b y ) z 2  2  2  2  - 0.  The d i v i s o r 0 i s defined by the id«al "a x  0"  y &  Let F be the d i v i s o r d e f i n e d by (j/  bifihe  * mA  a iipjSiri§ f  •» b y 2  b x 2  2  - xO  i  + (y  kinii?i*)&2z=£=&  2  + 3byz + M%»U*v)  u » [ t x - (**/b) ]t 2  3b z )0 2  2  mm foM'Stii'i  v » Ctx  (g/b) ] 2  3b 2. 2  3  + 1  B  It i a easily v e r i f i e d x -  + °M  y - [tx - ( a / b ) ] x  2  b t 2  that z -  2  - (-;/ -) J ;  2  3  t X  3b y  2  2  °" d e f i n e s an i n v o l u t i o n of k(x,y,z) which gives a b i r a t i o n a l transformation  of V i n t o V.  The  of t h i s mapping can  nature  more c l e a r l y understood by observing  that, i n vector notation,  y- l ( t , u , v )  c(x,y,z)  be  Q^cek.  c Let V 'denote the surface i n ( t , u , v ) - s p « c 9 which i s biration<?.lly 0  e q u i v a l e n t to V, i . e . V t  3  + u  3  cr  + 3(at  2  i s defined  by  + bu )v + 3(a t + b u)v* » 2  2  2  0.  cr Similarly D  and  F  are obtained  by r e p l a c i n g x,y,z  r e s p e c t i v e l y i n the d e f i n i n g equations  f o r D and  F.  by P  t,u,v i s th©  vertex of the cone V . We  s h a l l need the f o l l o w i n g p r o p e r t i e s of V which Nagwta proves  in  [7].  F i r s t we  Proposition 1  give a general p r o p o e i t l o n ,  Let 0  be the d i v i s o r on a normal a f f i n e cone \J  d e f i n e d by a hettelSftfitffi J>tUai& id * ^wA«cipal on V i f ,  and  tfifrii §  &•  only i f , i t As. l o c a l l y  p r i n c i p a l at the vertex P  of V .  9  Proofs  Neceaaity i s obvious.  For s u f f i c i e n c y , assume that  i s l o c a l l y principal at P . element f C Op  such that  the l e a d i n g form o f f . ie a unit i n  Since  f  .  3  Then there e x i s t s an  < ^  m  *  Then f £c£  * .  %  L  8  *  °e  Therefore f ' / f  Hence  i s homogeneous, f  g e n e r a t e s ^ , and ^  is  p r i n c i p a l on V .  We apply  P r o p o s i t i o n 1 to the f o l l o w i n g lemma which p e r t a i n s  s p e c i f i c a l l y t o the cone V ( V ) as defined  Lemma 1  For any n a t u r a l number n, nD i s not p r i n c i p a l l o c a l l y at the vertex  Proofs  above.  P of V .  L e t E* be the p o i n t  (1,-1,0) on V * .  Now assume t h a t  f o r some n a t u r a l number n, nD i s p r i n c i p a l l o c a l l y at P.  Then by P r o p o s i t i o n 1, nD i s p r i n c i p a l on V . L e t  f be the homogeneous form which d e f i n e s nD. the degree o f f .  L e t m be  Then f / ( x + y ) i s a f u n c t i o n on V * m  whose zero and pole are nD* and 3mE* r e s p e c t i v e l y , and we have nD* - 3mE* ^-0,  hence n(D* - E * ) ^ 0  (3m » n ) ,  which i s a c o n t r a d i c t i o n because E* i s r a t i o n a l over R and  D* i s a generic p o i n t o f V* over R.  10  We remark here that the r e s t r i c t i o n i s not necessary.  that n be a p o s i t i v e i n t e g e r  An obvious m o d i f i c a t i o n o f the proof  us t h s r e s u l t f o r any non-zero i n t e g e r n.  Lemma %  k [ x,y,z,t,u,v]  a k[x,z,t,v]  gives  F i n a l l y we show  d e f i n e s the a f f i n e  model V - F. Prooft  L e t A be the a f f i n e v a r i e t y d e f i n e d  by k[x,y,z,t} and  l e t A* be the a f f i n e v a r i e t y d e f i n e d Since  a x + 2  b y€.d  t£0(" , i.e. 2  and  2  and s i n c e x ^ *  0[ \ ^0', 2  <  Since  dn^ ,  Ct0-[t\  2  3  - I f (y/x)  3  - -3 [ b y z + a x z + ( a x + b y ) z 2  2  2  2  we see t h a t x v € O" and o b v i o u s l y f  f o r A* the same property, good.  - (x + y )/x 3  2  ]/x  3  3  zv € 0~[t\ .  3  , Therefore  as s t a t e d above f o r A, holds  F i n a l l y the d e f i n i t i o n o f v shows t h a t  x^[tf l v  Proofs  i s generated by x  Since  1 • (tx - ( a / b ) )  Corollary  we see t h a t  z , A - ( d i v i s o r d e f i n e d by x « z • 0 ) c o i n c i d e s  with V - F.  and  by k [ x , y , z , t , v ] . '  + z^[t,v]  contains  1.  Therefore  A* « V - F„  the a s s e r t i o n i s proved.  V - F » V  Sincev  - F  i s an i n v o l u t i o n , Lemma 2 gives us the r e s u l t .  11  Lemma 3  ^  p  H  ^ . k  We s t a t e Lemma 3 without proof p a r t l y because the proof i s rather involved  and p a r t l y because the f i r s t  h a l f o f the proof  of Theorem 2 i n § 4 can be adapted t o a proof o f Lemma 3.  Nagata proves that M • V U V  i s normal but not p r o j e c t i v e .  contend that M a l s o i s not d i v i s o r i a l . some f u r t h e r p r o p e r t i e s  of i t .  but f i r s t we s h a l l  We prove  12  4  PRELIMINARY RESULTS  V i s d e f i n e d by x  + y  3  + 3(ax  3  + by )z + 3(a x + b y ) z  2  2  2  2  We i n t r o d u c e homogeneous c o o r d i n a t e s x^ x • ] / 4 » y " 2^ 4 x  x  x  »  x  " 3^ 4  2  x  '  x  a n  ^  w  - 0  2  ,x^  (I)  where  °btain a r e p r e s e n t a -  e  t i o n o f V i n terms o f homogeneous c o o r d i n a t e s * x  3 x  + x + 3(ax + b x 3  2  2  x  2 2  )x  + 3(a x  + b x )x  2  3  2  x  2  2  3  « 0  S i m i l a r l y i f we i n t r o d u c e homogeneous c o o r d i n a t e s y^ , y y  4  where t • Y j / y ^ •  » ^2^A  u  *  ™ 3^ 4  v  y  »  y  w  o  O D  '  t a i n  D O  *  (IN »y »  2  3  h  non-homogeneous and homogeneous r e p r e s e n t a t i o n s o f v"" i t' + u y  3 x  + y  + 3(at  3  3 2  +  3(a  + bu )v + 3(a t + b u ) v  2  2  2 yjL  + by  We s h a l l study more deeply  2  2 2  )y  2  + 3(a y 2  3  - 0  2  + b y )y 2  x  2  2 3  (II) -  0  (II»)  the b i r a t i o n a l t r a n s f o r m a t i o n between  the two p r o j e c t i v e cones, V and V , the c l o s u r e s o f V and V r  r  being taken i n x and y space r e s p e c t i v e l y where x • ( x ^ , x , x , x ^ ) 2  and y « (yj^y^y-j.y^)»  Let ^ i f  denote the l o c a l r i n g i n  k ( x , y z ) o f the l i n e at i n f i n i t y o f V , p  H  m (y^,y ,y 0), l e t d e n o t e 2  3 >  Similarly l e t ^  p  3  If ^ £ L^ , i.e. i f  i t s l o c a l ring i n k(t.u.v).  denote the l o c a l r i n g i n k(x,y,z) o f P.  13  <^  Lemma 1 Prooft  <^_<r  c p  for a l l %  We f i r s t show that  L ^ where  c  { x,y,zj «n^  ^VF^. , the maximal i d e a l  c  r  of ^ L i f . x - a t + b u » a (y /y ) + b (y /y ) - a y y 2  2  2  2  1  b t 2  2  4  2  b (  2  2  /y J  y i  4  x  b  2  4  + b y y 2  4  2  2  4  2 y j L  Similarly y • (y /yi)x 2  " L*x - (a/b) ] -.- 1 - ( y / y ^  Z  2  3  3b v  3  by  2 2  )y  3b  y  a  y  l  2  +  by  2  >  (  <  y  3 y i  2"" u2 3 l 2  + 3(a  3  >  2  y  2  y i  +  2  4  + b y )y  2  y i  3b y  2  - - y ^ U y ^ *  3  4  3b (y /y )  2  " 4[  + 1 - y (  3  2  2 3  3 x  y y  3 2  )  3  ]  3  l  *  b 2 y  2  ) y  3^  by  Hence since y^ + 0 and y k Cx»y» 1 z  c  6fr  1 1  , [ x , y , z ) c m ^ and  ^° complete the proof, i t s u f f i c e s to show that  *  €  _ i _ € . ^ r  V  r  g(x,y,z) Since  • Q for ^  4  g(x,y,z)  T  ^ p  g(0,0,0) <^ 0}  Q  hence g(x,y,z) • f(x,y„z) • e  where every term i n f c o n t a i n s e i t h e r x or y o r z« 1  m  g(x,y,z) where  1 c • f(x,y»z)  V (y > - ( b y 2  A  a  1  2  )  B ,  (b y 2  1  3  1 c ^(y ) + x  ) (b y n  2  3 1  )  p  Ffy^y^y^y^  and F - f •  £ y k[y] , 4  14  i . e . we have c l e a r e d denominators i n f . Since  - _1_,  c ^(y ) + ny^yg.yg.Q)  c  1  1 '  c &  %  g(x,y,z)  We r e s t a t e t h i s r e s u l t aa  Apart from F n L  Corollary 1  <r  c.tr  f €.  Corollary 2 Proofs  ^  Take ^  Lemma 2  $« % The  '  i s not fundamental.  generic  a  , the l o c a l r i n g o f any p o i n t  ^ <r  c p  p  M  f o r L^>.  f o r a l l | £ L ^ where  % 4- F.  proof i s e x a c t l y s i m i l a r t o that o f Lemma 1.  Analogous c o r o l l a r i e s may be s t a t e d .  F  i s defined by the equations  t «  0  2 2 2 I u + 3buv • 3b v - 0 * We a s s e r t that F aa L^  and L  2  •  c o n s i s t s o f two l i n e s which we s h a l l denote T h i s may be Been as f o l l o w s .  Dehomogeniza *  15  at v to obtain  (u/v) + 3b(u/v) + 3 b - 0 2  2  I t f o l l o w s that m -3b + / 9 b - 1 2 b = •  u  2  v  Habca F.or « L^U l>  2  where  2  b (-3 ± 1 ^ / 3 ) 2  i s d e f i n e d by the equations  t - 0  u - (b/2) (-3 - i sl~3)v m 0 and L  i s defined by  2  ( t - 0 u - (b/2)(-3  +i 4 3)v - 0  ( 0 , l , 3 _ + 4 i 3 ,0) and L ^ L ^ - P 6b c  (0.1.-3 -i>/3 , 0 ) . I f we dehomogenize a t y , so t h a t 6b Y2  ^2^3  m  * ^3 " 4 ^ 3 y  y  *  w  e  0 D  * * a  n  non-homogeneous  m ^1/^3  »  coordinates  f o r these p o i n t s . (0, - b (3 +ivf3), 0) and P 2 A s i m i l a r d i s c u s s i o n holds L 0 L^ . 2  X  f o r F « L^U i - , 2  - (0, - b (3 -i,/3). 0) 2 - L^O L ^ , P j •  I f we dehomogenize ( I * ) at Xg and denote  2 " 2^*3 ' 3 " *4^*3 ' ° K  2  X  W  o  b  t  a  i  - j /. 3 > x  x  L  n  U - | ( X , X , X ) j X + X + 3(aX + bXj ) + 3 ( 8 ^ • b X ) 3  1  2  3  x  3  2  2  x  2  2  2  16  Let  p  Jf(x x x ) 1>  2t  3  i " I g(x X2,x ) lf  *?2 ' 2 €  If  L  w  h  e  r  3  ^2 *  e  9  l  B  ^ to- - 1 g(t,u,v)  g(Q, - b(3 + i / 3 ) , 0 ) + 0 2  t  g(0,u,v) f 0 where u •  b (-3 +i/3)v 2  We may now s t a t e Lemma 3 Proof 1  ^  c p  ^ | f  for a l l  We show f i r s t  L^  x  3  x  2  x  Similarly X  + t  3  u  - 3uv(a t + b u) u  2  + t  3  2  + l)/3b > 2  2  \  2  2  3  2  3tv(a t + b u)  2  3  P  2  ( (u/t)  3^ 4  2  2  * '  2  (a t + b u)/b t  x  3v(a t + b u) t /u  %  that { X ^ X j , X ^ c k(t,u,v)  X^ • X j ^ » j / 4 x  where  2  3^.3 + t  and X  - 3b t v 2  3  u  3  3  Hence  3  +t  3  f X , X , X } c k(t,u,v) and " [ X j ^ X ^ X ^ e k ( t , u , v ) . 1  2  to show that  1  g(X.,X-,X,) 'i»"2 "3  P  9  Let  I t suffices  3  l  >  1 S^fc: o(X X ,X ) l 9  2  3  Then g(X ,X ,X ) JL  2  3  gU^X^Xg) « (X + _ | J 3 2  + i^3) ) h ( X ) + f ( X X , X ) * 2  l f  2  3  17  where c + 0 and a l l terms i n f U ^ X ^ X ^  9  ( X  1' 2» 3 X  X  contain  either *  3  (^3uv(a t+b u) + b (3+i\l3)\htX.,) + f(X.,X-,X,) V 3 3 2 J  )  2  2  1  2  3  + t  (u [3uv(a t+b u)  + c  2  2  u  2  or X «  x  3  + t  3  )  m  (.lAtVf  + |(3+iV3)(u +t )][u +t ] " h(X ) + 3  3  3  3  m  1  2  ( X ^ X ^ )  + c(u +t )" 3  where m i s s e l e c t e d l a r g e enough ao t h a t [ u + 3  and (u + t ) f ( X , X , X ) € k [ t u , v ] . 1  2  3  t J " * h ( X 2 ) £ k[t,u,v] 3  m  To s i m p l i f y the  r o  f  3  notation,  let »  1  G(t,u,v)  9 v i»^2 3^ x  ,X  Let t • 0 and u -  b (-3 + iv/3)v, i . e . v - -1 (3 + i / 3 ) u . 2 6b  6(0,u,v) -  u  3 m  [3u( sJL-fa+iJa^ u ) b u + 6b 2  L  b (3+iJ3)u 1u 2 J 3  3 ( , ,  " h + 1 )  3ns  u  f-b -2  (34-i 3 ) u  3  •  b (3+i 3 ) u l u 2 J 3  L  m  i c  Hence G(t,u,v) «  • 1 g(X ,X ,X ) 1  2  3  Then  ^2  •  3 ( m  -  1 )  h + cu * 3  eu  3 m  is  Corollary 1  Apart from P , f o r a l l  L- ,  to- i s not  fundamental.  Corollary 2  ^  Lemma 4  ^ p 0  ^, ^ 2  C p r  l  u  ^  particular,  for a l l ^ ^ .,C€LL, ^ where where ? ^  p  r  2  1  44PP, , and i n  ^ $''.a" L  l  The proof i s e x a c t l y s i m i l a r t o that o f Lemma 3.  The preceding  lemmas e s t a b l i s h some o f the p r o p e r t i e s o f the  b i r a t i o n a l correspondence "~.  We now turn our a t t e n t i o n to the  behavior o f C a r t i e r d i v i s o r s on V.  Since V i s a b s o l u t e l y  d u e i b l e , we s h a l l work now over k « R(a,b) r a t h e r than It. aim  irre Wa  t o prove  Theorem 1  Suppose f(x,y,z}£ k[x,y,z] i s such that f ( P ) «• 0. Then i t cannot happen that ( f ) y f*\ v  f P^,P |.  The proof o f Theorem 1 w i l l r e q u i r e two lemmas. we develop a convenient  2  First  notation.  Let tTi be the unique maximal i d e a l o f &~ 0 k[x,y,z]» and l e t p  1?  f&7TI.  W r i t e f as a homogeneous polynomial  I n x , x , x , x, I  and  dehomogenize a t x .  '  The r e s u l t w i l l be a r a t i o n a l f u n c t i o n  3  i n X,, X^, X^ w i t h only X  Let  3  2.  i n the denominator.  3  L be d e f i n e d by ) z = 0  L x + y = 0 Let C be d e f i n e d b y f z = 0  - xy + y " = 0 2  Lemma 5  3L i s l o c a l l y p r i n c i p a l a t P.  3  3  Proof:  We prove t h i s on the o r i g i n a l cone X L  + Y  3 = Z .  i s then d e f i n e d by f Z = 0 X + Y =  X + Y =  Z 2. •  X-XY  -.  3  + Y  2.  0  i s a u n i t e f t L.  1 z X - X Y  + Y  2  3  Hence ord X + Y = 3. L .  The equation X  3  + Y  3  = Z  shows that X + Y has no other zeros on the cone. Hence 3L i s l o c a l l y p r i n c i p a l a t P.  20,  Lemma 6 The  3C i s l o c a l l y p r i n c i p a l a t P.  proof  i s e x a c t l y analogous t o that of Lemma $,  We r e t u r n t o the proof of Theorem 1 and proceed by c o n t r a diction.  L e t g ( X , , X , X ) be the numerator of the r a t i o n a l £  function  i n X, , X , X a  3  3  obtained  from f .  Our assumption  t h a t g(X, ,X£,0) i s a polynomial w i t h only P, and P Since g(X  (  ,X ,0) does not depend on X a  3  a  implies  as roots.  and the o r i g i n a l  equation of the cone i n these v a r i a b l e s becomes a c y l i n d e r independent of X  3  and g i s a r e g u l a r f u n c t i o n on U, we have  (g)  _ = nF - mL - pC V,V  n,m,p€-2-  Therefore (g ) 3  By  _ = 3nF - m3L - p3C . V,V  Lemmas $ and 6 , 3 L a n d 3C a r e l o c a l l y p r i n c i p a l , a n d t h i s  leads  t o a c o n t r a d i c t i o n of Lemma T i n § 3 ,  21  This result,  together w i t h the f a c t that  i n d i c a t e s s t r o n g l y that M = Vu V  r  2),  (lemma  i s not d i v i s o r i a l .  U n f o r t u n a t e l y Theorem 1 does not preclude the p o s s i b i l i t y t h a t ( f ) _ L = { P | ,P } P,V  and t h a t an i r r e d u c i b l e  r |  0 o  E of ( f ) V,V  ?  which does not pass through P,  a p o i n t P# d i f f e r e n t  from P, and  component  intersects  at  ? . z  To overcome t h i s d i f f i c u l t y , we must develop an a l g e b r a i c tool  of some power.  we  quote two  by  ZarIski-Samuel.  B e f o r e we attempt  the a c t u a l development,  lemmas, the f i r s t by Seidenb erg -Cohen, tke  Seidenb erg's Lemma  L e t R be  integrally closed  quotient  r i n g F.  in i t s total  I f f ( X ) and g ( X ) a r e  monic polynomials i n F[x]and is  i n R[x],  R [ X ] .  second  h(X)  = f  then f ( X ) and g ( X ) a r e i n  (X)g(X)  22  For the proof, see [6~\,  Z a r i s k i a Lemma 1  page  256.  Let T\ be a noetheriah domain and l e t Yi ' • TlCTf]  where T i s a l g e b r a i c over Vl be a domain which c o n t a i n s h i s a simple r i n g extension of Yl . 71',  Let 1  For the proof, see [ 2 ] , V o l . I I , page  Theorem 2  Then h{  i h( p j . ,  323.  Let £ be en i r r e d u c i b l e s u b v a r i e t y of V auch t h a t P^E.  Let tff be the (prime) i d e a l of k [ x , z , t , v ] .  which d e f i n e s E on V - F. by one Proofs  be a prime i d e a l i n  and l e t fl » p , 0 Vl .  different f r o m V  and  Then Of i s generated  element q, and Var(q)y • E.  Since P^-E,  Gf c o n t a i n s an element f ( x , y , z ) such that  f(P) « 1, i . e . f ( x , y , z ) • g(x,y,z) no constant term.  + 1 where g c o n t a i n s  Let § be the i n t e g r a l c l o s u r e of  k£tx,tz^ i n k ( t x , t z ) .  Since tx and t z are f u n c t i o n s  on the cubic curve V*,  we  1 over k and  see that § i s o f  dimension  hence t i s t r a n s c e n d e n t a l over S .  It i s  e a s i l y seen t h a t f o r a s u f f i c i e n t l y l a r g e i n t e g e r n, t f n  i s a monic polynomial i n t with c o e f f i c i e n t s i n k[tx,tz~\s hence a f o r t i o r i t " f i s monic i n t with in  S.  t f£ n  coefficients  <n J\>3.  Since ,1[t] i s n o e t h e r i a n , there e x i s t s at l e a s t  one  23  prime may  f a c t o r i z a t i o n o f t f » q^. . ^  *  m<  be chosen monic i n t .  One  m  which a l l the f a c t o r s  n  of these f a c t o r s , say q,  must be an element of.61 , otherwise 0? would not be  Let TQ be the q u o t i e n t f i e l d of ^ .  We  assert that q i s i r -  r e d u c i b l e i n I ^ L t ] , f o r otherwise l e t q • <? (t) f a c t o r i z a t i o n of q i n t o polynomials i n ^ [ t l * polynomials, of course, may Lemma, cf(t) and  *(t)€J[t],  b i l i t y of q i n ^ [ t l .  be taken monic.  be a  By  Seidenberg'a the i r r e d u c i -  i s f a c t o r i a l , the i d e a l  generated by q i n  vfgLt~\, ( q ) * , i s prime.  i s a prime i d e a l .  Clearly ( q ) c O l n J[t],  Consider the diagram  *Mt)  Both these  thus c o n t r a d i c t i n g  Since vT^t^  prime.  ( f o l l o w i n g page)t  Hence (q) •  (q)*r\i[tl  24  0 r  We note f i r s t that k [ x , z , t , v , l / t , l / x l  - j[t,l/t,l/tx] .  This  f o l l o w s from the c o n s i d e r a t i o n s ! k[x,z,t,v,l/t,l/xl £  k[tx,tz,t,l/t,l/tx]  The only d i f f i c u l t y i n proving t h i s i n e q u a l i t y i s to show vck[tx»tz t l/t»l/txl B  "  9  and t h i s may  9  C t x - (a/b) 3 2  3  be seen from the f o l l o w i n g !  + 1 . (v/x)  3  2 3b z  3b z 2  + 1 -  ^2 3 b x z r 2 2 2 2 i - Lby + i x + (a x + b y ) z J b x 2  3  3  + v  3b x z  - [ b y * z + ax*z + (a*x + b y ) z ] 2  x  2  3  25  Hence k [ x z , t , v , l / t , l / x 1 £ ( f [ t , l / t , l / t x l . f  k [ t x t z ] S k[x,z,t,v,l/t,l/x3 f  k [ x , z , t , v , l / t 3 i s the coordinate variety V  - Itjy*' -  Clearly  .  r i n g o f the  non-singular  hence i t i s i n t e g r a l l y c l o s e d .  <r  v  Thus  f £  k[x,z,t,v,l/t].  and  t h i s e s t a b l i s h e s the e q u a l i t y k [ x , z , t , v , l / t , l / x ] » i f [ t , l / t , l / t x 3 .  C l e a r l y \f [t, l / t , 1/tx 3 ^ k [x,z, t, v , l / t , l / x 3 ,  t and x are prime elements i n k [ x , z , t , v ] s i n c e d e f i n e s the d i v i s o r D and t « x . element i n "7 • is  Also n e i t h e r t nor x i s an  Hence <rf , the extension 8  a prime i d e a l .  xk[x,z,t,v]  of  to k [x,z, t , v , l / t , l / x l ,  S i m i l a r l y ( q ) , the extension 8  I[ t , l / t , l / t x l , i s prime.  f  » ^ n B  kTx,z,t,v3.  o f (q) i n Hence a  double a p p l i c a t i o n o f Z a r i s k i ' s Lemma gives us that h ( < ^ ) ^ h("7)f 8  however, f i s a minimal i d e a l , i . e . h(<f ) « 1. s i n c e E i s o f codimension 1. (q)  e  Therefore  h (  8  ) • 1.  i e prime, we see t h a t ( q ) « e  by one element q & ^ o  Since  ( q ) e <tf  Hence f  8  8  B  and  i s generated  follows that * f i s l i k e w i s e . p r i n c i p a l .  To complete the proof o f Theorem 2, we observe, using our double notation  f o r d i v i s o r s , that we have ^V-F,M "  Hence  E  '  26  (q)y ^ • E + mF P<£E=^-  (q)  p  M  where m €. Z  « roF ; however, Lemma 1 i n  that m • 0, otherwise mF fore f o l l o w s that (q)w  We  M  i s l o c a l l y p r i n c i p a l at P .  •.  Corollary 1  q€k[x y z]  Corollary 2  q(P)  t  t  + 0 .  us  I t there-  • E.  d e r i v e from the f a c t that (q)y ^ -  corollariest  f 3 then gives  E the f o l l o w i n g immediate  27  5  MAIN THEOREM  With Theorem 2, we are i n the p o s i t i o n to proves  Theorem 3 Proofs /  M «• V U V ^ i s not d i v i s o r i a l .  Assume to the contrary that i t i s .  Then there  exists  a p o s i t i v e C a r t i e r d i v i s o r D o f M such that P €. VarD and P°V Var D. that 0  P,M - ( f )P,M *  f£ k[x,y,z].  P^E  i  L e t f be the r a t i o n a l f u n c t i o n We may assume, i n f a c t ,  f o r a l l i , where the E^ are i r r e d u c i b l e .  g(P) t  The  </ i n kCV-F] which d e f i n e the E. are r  ±  p r i n c i p a l by Theorem 2.  L e t q. generate  0  Consider now < /a>V-F.M f  that  Let  (prima) i d e a l s  (ii)  such  if) V-F,M - (g)V-F,M  *f, and d e f i n e  28  -  (f)  p,M +  " P,M D  +  n  -nF  E  "  p  (f/g) _ v  »  F > M  I t then f o l l o w s (f/g) <J" V  m  E  +  n  F  T does not appear i n  + nF, the purpose o f the nF being component i n Dp ^ .  H i i  here only the non-empty i n t e r s e c -  ( f / g ) ^ with V - F  t i o n o f the d i v i s o r M  m  F  Note t h a t s i n c e we c o n s i d e r  Dp  2I i i  to c a n c e l a p o s s i b l e  By Lemma 2 i n $ 3,  (tVg) <r_ <r v  F  > M  that M  • Dp p| • nF + m F  r  wherera€.  i f ,  By assumption, P j- Var D, hence a f o r t i o r i P <f Dp ^ .  We  have ( f / g ) p ^ p , » raF^ T h i s , however, c o n t r a d i c t s Lemma 1 i n $ 3 unless m • 0. i f m - 0, we have that f / g ^ p * ' , but s i n c e g(P) f/g^^p  •  Then  M 0 (ii),  Hence  whence i t f o l l o w s from Lemma 3 i n § 3 that f / g i s a constant. T h i s constant  must be zero  a contradiction.  since  f (P) • 0, and t h i s i s c l e a r l y g  29  Note that Theorem 3 provides i n independent proof of Nagata's contention, i . e .  Corollary Proofs  M cannot be imbedded i n p r o j e c t i v e  Assume to the contrary that i t can. d i v i s o r i a l , contradicting M i s not p r o j e c t i v e .  space. Then M must be  the previous theorem*  Hence  30  6  CONCLUSION  We  o f f e r an o b s e r v a t i o n concerning the polynomial theorem of  Snapper which B o r e l l i as  extended i n [5] to d i v i s o r i a l v a r i e t i e s  followss  v a r i e t y and "X an a d d i t i v e sheaf f u n c t i o n ,  Let X be a d i v i s o r i a l i.e,  a f u n c t i o n defined  on the category of sheaves with valuaa  i n an a r b i t r a r y a b e l i a n group, say H  , and such that f o r every  exact sequence 0 /UF)  V F•  m A(F')  v F  % (F ). M  F••  — — y  0  Then f o r every aheaf F over X and  every f i n i t e sat of l i n e c l a s s e s f ^ , . . . , f m« aion A Tf^ degree  of X. the expres-  m . . .f  n n  ( F ) J i s a polynomial i n m^,  . . ..m^  of  ^ dim(Supp F ) .  Although M i s not d i v i s o r i a l , satisfies  the question asises whether i t  t h i s extension of Snapper a 8  Theorem.  I t has r e c e n t l y  been brought to our a t t e n t i o n that t h i s question has been answered i n the a f f i r m a t i v e by Kleinman  f o r the case where J « X,  the  E u l e r c h a r a c t e r i s t i c , and h i s proof extends to the case o f an a r b i t r a r y a d d i t i v e sheaf f u n c t i o n .  31  BIBLIOGRAPHY  Serge Lang. I n t r o d u c t i o n to A l g e b r a i c Geometry. science P u b l i s h e r s , Inc., New Ys»»k.  Inter-  Oscar Z a r i s k i and P i e r r e Samuel. Commutative Algebra,. V o l s . I&.II, Van Nostrand, P r i n c e t o n . New J e r s e y . Jean-Pierre Serre. "Faisceaux Algebriquep Coherent^", Annals o f Mathematica, v.61, 1955, pp.197-278. Ernst Snapper. " M u l t i p l e s o f d i v i s o r s " , J o u r n a l o f Mathematics and Mechanics, t.8 (1959), pp.967-992. Mario B o r e l l i . " D i v i a o r i a l V a r i e t i e s " , P a c i f i c Journal of Mathematics, Vol,13, No.a, 1963, pp. 375*386. i  S. Ssidenberg and 1.5. Cohan. "Prime Ideals emd I n t e g r a l Dependence", B u l l e t i n of American Mathematical 5ociety 52, 1946, pp. 252-261, Masayoahi Nagata. "On the i m b e d d i n g of a b s t r a c t s u r faces i n p r o j e c t i v e v a r i e t i e s " . Memoirs of College of Science, U n i v e r s i t y o f Kyoto, S e r i e s A , Vol.XXX, Mathematica No.3, 1957, pp. 231-235.