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UBC Theses and Dissertations

The Schubert calculus Higham, David Paul 1979

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THE SCHUBERT CALCULUS by DAVID PAUL HIGHAM B.Sc,  Mount A l l i s o n U n i v e r s i t y , 1973  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department o f Mathematics  We accept t h i s t h e s i s as conforming to t h e r e q u i r e d s t a n d a r d  THE UNIVERSITY OF BRITISH COLUMBIA October 1979  D a v i d P a u l Higham, 1979  In  presenting  an  advanced  the I  Library  further  for  this  thesis  degree shall  agree  scholarly  at the University make  that  purposes  h i s representatives.  of  this  written  for financial  of  University  Date  gain  Columbia  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  October 1, 1979  of British  Columbia,  f o r reference copying  by t h e Head  i s understood  Mathematics of British  o f the requirements  for extensive  may b e g r a n t e d It  fulfilment  available  permission.  Department The  it freely  permission  by  thesis  in partial  shall  that  I agree  for that  and study.  of this  thesis  o f my D e p a r t m e n t o r  copying  n o t be a l l o w e d  or  publication  w i t h o u t my  ii ABSTRACT An enumerative problem asks the f o l l o w i n g type of q u e s t i o n ; many f i g u r e s ( l i n e s , p l a n e s , c o n i e s ,  c u b i c s , e t c . ) meet  how  transversely  (or a r e tangent t o ) a c e r t a i n number of o t h e r f i g u r e s i n g e n e r a l position?  The l a s t c e n t u r y saw the development  of a c a l c u l u s f o r s o l v i n g  t h i s problem and a l a r g e number of examples were worked out by S c h u b e r t , a f t e r whom the c a l c u l u s i s named. The c a l c u l u s , however, was not r i g o r o u s l y j u s t i f i e d , most e s p e c i a l l y i t s main p r i n c i p l e whose modern i n t e r p r e t a t i o n i s t h a t when of an enumerative problem a r e v a r i e d c o n t i n u o u s l y  conditions  then the number of  s o l u t i o n s i n the g e n e r a l case i s the same as the number of s o l u t i o n s i n the s p e c i a l case counted w i t h m u l t i p l i c i t i e s . p r i n c i p l e of c o n s e r v a t i o n  Schubert c a l l e d i t the  of number.  To d a t e the p r i n c i p l e has been v a l i d a t e d i n the case where the f i g u r e s are l i n e a r spaces i n complex p r o j e c t i v e space, but o n l y i s o l a t e d cases have been s o l v e d where the f i g u r e s a r e c u r v e d .  H i l b e r t c o n s i d e r e d the  Schubert c a l c u l u s of s u f f i c i e n t importance t o r e q u e s t i t s j u s t i f i c a t i o n i n h i s f i f t e e n t h problem. We t r a c e the f i r s t f o u n d a t i o n of the c a l c u l u s due p r i m a r i l y to Lefschetz,  van der Waerden and Ehresmann.  b e i n g a summary of Kleiman's e x p o s i t o r y problem.  We d e s c r i b e  f o r m a l l y and d e s c r i b e  the Grassmannian  The i n t r o d u c t i o n i s h i s t o r i c a l ,  a r t i c l e on H i l b e r t -s f i f t e e n t h and i t s Schubert s u b v a r i e t i e s more  e x p l i c i t l y the homology of the Grassmannian  g i v e s a f o u n d a t i o n f o r t h e c a l c u l u s i n terms of a l g e b r a i c c y c l e s . we compute two examples and b r i e f l y mention some more r e c e n t  which Finally  developments.  iii TABLE OF CONTENTS ABSTRACT  .  TABLE OF CONTENTS LIST OF FIGURES  i i i i i , , . . .  iv  ACKNOWLEDGEMENT  v  Introduction  1  Chapter I  THE GRASSMANNIAN  §1  The Naked Grassmannian  13  §2  The Grassmannian as V a r i e t y  16  §3  The Grassmannian as M a n i f o l d  22  §4  The U n i v e r s a l Bundle over t h e Grassmannian  24  §5  The Dual Grassmannian  26  Chapter I I THE SCHUBERT VARIETIES §1  The D e f i n i t i o n  28  §2  Example: The Schubert V a r i e t i e s i n G^E^)  33  Chapter I I I THE SCHUBERT CALCULUS §1  I n t e r s e c t i o n Theory  39  §2  The Grassmannian as C.W. Complex  46  §3  The R i n g S t r u c t u r e i n Homology  51  Chapter IV  MORE RECENT DEVELOPMENTS  §1  The Hasse Diagram  60  §2  C o n c l u d i n g Remarks  72  BIBLIOGRAPHY  76  iv LIST OF FIGURES  ^2  Figure 1  3  Figure 2  ^  Figure 3 Figure 4 Figure 5 Figure 6  6 1  6 1  ^2  h  3  5  ^  2  5  6 1  H  R  H  Q  a  6 1  62  V  ACKNOWLEDGEMENT I am i n d e b t e d for h i s patience pletion.  m a i n l y t o L a r r y R o b e r t s , my a d v i s o r ,  i n s e e i n g t h i s work through t o i t s com-  I would a l s o l i k e t o thank J i m C a r r e l l f o r t h e  o r i g i n a l i d e a and f o r many o f the u s e f u l  references.  G r a t i t u d e i s a l s o due t o Roy D o u g l a s , Mark Goresky, J i m L e w i s , Ron R i d d e l l and B i l l Symes f o r t h e i r and  sometimes i n s p i r i n g  sympathetic  discussions.  Worthy o f mention a l s o a r e those who h e l p e d i n a non-professional  c a p a c i t y by p r o v i d i n g encouragement,  moral s u p p o r t , t e a and sympathy.  Those uppermost i n my  mind a r e Roy Douglas, Ed G r a n i r e r , Fred Henry, Mike Ken  Margolick,  S t r a i t o n and S c o t t Sudbeck. F i n a l l y , a word o f thanks t o the " b e h i n d - t h e - s c e n e s "  people Mrs. MacDonald and Kathy Agnew f o r t h e i r many k i n d n e s s e s and b u r e a u c r a t i c  s h o r t - c u t s , and Mrs. Janet  C l a r k f o r h e r i n t e l l i g e n t t y p i n g of t h e m a n u s c r i p t .  - 1 -  INTRODUCTION D u r i n g the l a s t c e n t u r y work i n geometry was h i g h l y i n t u i t i v e . was  e s p e c i a l l y t r u e of the so c a l l e d enumerative geometry, which  t o answer the q u e s t i o n "How s e t of g e o m e t r i c  This  attempted  many f i g u r e s i n g e n e r a l s a t i s f y a p r e s c r i b e d  c o n d i t i o n s ? " A s i m p l e example of t h i s i s to f i n d  the  number of l i n e s t h a t meet f o u r g i v e n l i n e s i n g e n e r a l p o s i t i o n i n 3-space. P o n c e l e t began work on q u e s t i o n s o f t h i s n a t u r e w h i l e i n a R u s s i a n m i l i t a r y p r i s o n a t Saratow i n 1813.  He p u b l i s h e d a paper e n t i t l e d  des p r o p r i g t e s p r o j e c t i v e s des f i g u r e s i n 1822 n o t i o n c a l l e d the p r i n c i p l e of c o n t i n u i t y .  Traite  i n which he i n t r o d u c e d a  Roughly p u t , the p r i n c i p l e  s t a t e s t h a t the number of s o l u t i o n s t o an enumerative problem does not change i f the parameters are v a r i e d c o n t i n u o u s l y .  The p r i n c i p l e was  not  p r o p e r l y j u s t i f i e d , and Cauchy c r i t i c i z e d i t s e r i o u s l y b e f o r e the paper was  even p u b l i s h e d .  I n s p i t e of Cauchy's i n f l u e n c e , which c r e a t e d some  p r e j u d i c e , the p r i n c i p l e o b t a i n e d widespread  p o p u l a r i t y and t h e r e s u l t i n g  c o n t r o v e r s y has not been c o m p l e t e l y r e s o l v e d even t o t h i s v e r y Hermann Casar H a n n i b a l Schubert was  a p r o l i f i c geometer and,  r e v i v e d the p r i n c i p l e , used i t t o c a l c u l a t e the s o l u t i o n s t o an number of enumerative problems.  day. having astounding  H i s f e r t i l e mind produced numbers t h a t  were o f t e n i n t h e tens and hundreds of thousands or more, l o n g b e f o r e advent of t h e modern e l e c t r o n i c computer, though, i r o n i c a l l y , a f t e r development of t h e i l l - f a t e d I n 1874  Schubert  the  the  " d i f f e r e n c e e n g i n e " of C h a r l e s Babbage.  changed the name of the p r i n c i p l e to the p r i n c i p l e of  s p e c i a l p o s i t i o n i n an attempt  t o a v o i d the p r e j u d i c e .  was not s a t i s f i e d t h a t t h i s name embodied the n o t i o n of  S c h u b e r t , however, continuous  v a r i a t i o n , and so the p r i n c i p l e r e c e i v e d i t s f i n a l b a p t i s m , two y e a r s as the p r i n c i p l e of c o n s e r v a t i o n of number.  D e s p i t e the w e a l t h o f h i s  later,  - 2c o n t r i b u t i o n s t o enumerative geometry though, Schubert r e a l i z e d t h a t t h e p r i n c i p l e s t i l l needed t o be Returning  confirmed.  now t o t h e example mentioned above we w i l l see how Schubert  answered t h e q u e s t i o n .  L e t t h e f o u r l i n e s be L^, L^, L^ and L^ and assume  t h a t they a r e i n g e n e r a l p o s i t i o n .  Now move L^ so t h a t i t i n t e r s e c t s L^  at P , and move L^ so t h a t i t i n t e r s e c t s L^ a t Q .  The l i n e s a r e now i n  " s p e c i a l p o s i t i o n " and i t i s easy t o count t h e l i n e s t h a t pass through a l l these four l i n e s .  One l i n e , L, i s d e f i n e d by P and Q , and s i n c e each p a i r  of i n t e r s e c t i n g l i n e s spans a p l a n e , t h e l i n e o f i n t e r s e c t i o n L', o f t h e two p l a n e s i s a second l i n e p a s s i n g t h r o u g h a l l f o u r l i n e s i n s p e c i a l position.  Suppose t h e r e i s a t h i r d l i n e L".  To a v o i d n o t a t i o n a l  clumsiness  we w i l l denote t h e s p e c i a l i z e d l i n e s by L^ and L^ a l s o .  l e t R. be.the p o i n t o f i n t e r s e c t i o n o f L" and L.. 1  1  from L, L" does n o t pass through b o t h P and Q . P,  Now  Since. L" i s d i s t i n c t  Thus i f L" p a s s e s through  s a y , then as Q i s t h e o n l y p o i n t common t o L^ and L^, R^ and R^ a r e  distinct, i.e.  contains a t l e a s t three p o i n t s . equivalently { Q J R ^ J R ^ } ) .  F i r s t suppose t h a t A =  {P,R^,R^}  (or  The l i n e d e f i n e d by R ^ and R ^ ( i . e . L" i t s e l f )  l i e s i n t h e p l a n e spanned by L^ and L^, f o r c i n g P t o l i e i n t h a t p l a n e also.  But then t h e two p l a n e s must be c o i n c i d e n t , i n which case t h e r e  would be an i n f i n i t e number o f l i n e s p a s s i n g through t h e f o u r g i v e n Secondly suppose t h a t t h e R ^ a r e a l l d i s t i n c t ;  then L" l i e s i n b o t h  lines. planes,  and s i n c e L' and L" a r e d i s t i n c t t h i s a g a i n f o r c e s t h e two p l a n e s t o be coincident.  - 3 -  Schubert then brought into play the principle of conservation of number, which rested on a weak foundation, to conclude forthwith that the number of lines meeting a l l four given lines remains two when  and  are returned to general position, provided of course that the number i s f i n i t e i n the f i r s t place.  Incidentally, another degenerate case to avoid  is the p o s s i b i l i t y of a l l four lines meeting at a single point. The power of this technique was unmistakable. Schubert published his book Kalkul der abz'ahlenden Geometrie i n 1879 and i n i t he computed number after number of solutions to enumerative problems.  A l l the examples  calculated,. l i k e the one above, were in 3-space, but that did not prevent them from being extremely complicated. Witness the two sensational numbers of 666,841,048 quadric surfaces tangent to 9 given quadric surfaces, and 5,819,539,783,680 twisted cubic space curves tangent to 12 given quadric surfaces whose v a l i d i t y has s t i l l not been established.  Schubert later  worked i n higher dimensions. In 1886.Schubert obtained the number h!k!(k-1)!...3!2! n!(n-l)!...(n-k)! of k-planes i n n-space meeting h = (k+1)(n-k) general (n-k-1) planes. This number, however, has been found to be correct. The need to verify this principle i s best expressed :in the statement of Hilbert's fifteenth problem, the text of which, translated i n 1902 by Newson i s as follows: The problem consists i n this: To establish rigorously and with an exact determination of the limits of their v a l i d i t y those geometrical numbers which Schubert especially has determined on the basis, of the so-called principle of special position, or conservation of number, by means of the enumerative calculus developed by him.  -  4  -  A l t h o u g h the a l g e b r a of today g u a r a n t e e s , i n p r i n c i p l e ,  the  p o s s i b i l i t y of c a r r y i n g out the p r o c e s s e s of e l i m i n a t i o n , y e t for  the p r o o f of the theorems of enumerative geometry  decidedly  more i s r e q u i s i t e , namely, the a c t u a l c a r r y i n g out of the p r o c e s s of e l i m i n a t i o n i n the case of e q u a t i o n s of s p e c i a l form i n such a way  t h a t the degree of the f i n a l e q u a t i o n s and  the m u l t i p l i c i t y  of t h e i r s o l u t i o n s may  be  Poncelet  claimed, t h a t the p r i n c i p l e c o u l d be  had,  i n 1822,  foreseen. verified  a l g e b r a i c a l l y but d i d n ' t do so because he f e l t t h a t the problem should viewed p u r e l y g e o m e t r i c a l l y .  Schubert f e l t the same way,  though he  be  stated  i n h i s book t h a t i f the p r i n c i p l e were i n t e r p r e t e d a l g e b r a i c a l l y i t would amount t o s a y i n g t h a t the number of r o o t s of an e q u a t i o n the c o e f f i c i e n t s are v a r i e d .  E a r l i e r , i n 1866,  doesn't change i f  de J o n q u i e r e s had  e s t a b l i s h t h i s by a p p l y i n g the fundamental theorem of a l g e b r a . a polynomial  tried  But  since  can have r e p e a t e d r o o t s we might e x p e c t t o have m u l t i p l i c i t i e s  to contend w i t h sometimes., and example, we now  indeed such i s the case a s , r e t u r n i n g t o  p a r a l l e l but d i s t i n c t from the p l a n e of t h e r e i s no l i n e p a s s i n g  p o i n t s , and We  our  demonstrate.  Suppose a f t e r s p e c i a l i z i n g the l i n e s t h a t the p l a n e of L^, and  infinity.  to  and L^,  is  then r a t h e r than say  through a l l f o u r l i n e s , we  a l l o w the s o l u t i o n a t  To have the complete p i c t u r e we a l s o want to i n c l u d e i m a g i n a r y so the ambient space i s complex p r o j e c t i v e 3-space.  have r u l e d out the p o s s i b i l i t y of no s o l u t i o n but we  out the p o s s i b i l i t y of o n l y one Choose L^,  and  cannot r u l e  s o l u t i o n as the f o l l o w i n g w i l l show.  to be t h r e e skew l i n e s and  choose P ^ e L ^ .  TI^ be the span of P^ and L^ and 11^ be the span of P^ and L^. L^ do not i n t e r s e c t nor are they p a r a l l e l then H  0  and  are  Let  S i n c e L^ distinct.  and  Now  choose L, to be n . . n : l l - i 4 2 3  l e t P. = L. n L. f o r i = 1,2,3 i 4 1 '  are d i s t i n c t s i n c e the t h r e e l i n e s a r e skew. f o u r . l i n e s L ^ , so now L_^.  Since  and these p o i n t s  L, i s a l i n e t h a t meets a l l 4  assume that L i s a d i f f e r e n t l i n e p a s s i n g through a l l  i s determined by P.  and P^ L cannot pass through both  of  these p o i n t s , so assume, without l o s s o f g e n e r a l i t y , t h a t L does not pass through 7^.  L meets both  and  so l i e s i n the p l a n e d e f i n e d by them,  but s i n c e L. and L. both l i e i n n~ t h i s plane i s e x a c t l y n„• 2 4 2 2 (ji ^ L n P^.  '£112^1"^ = {P-j}  i . e . L meets  a t P^.  have  S i m i l a r l y L meets  But then L i s d e f i n e d by P^ and P^ and so must be  d i c t i n g the h y p o t h e s i s t h a t L and  Thus we  are d i s t i n c t .  itself,  Thus  at  contra-  i s the unique  l i n e meeting a l l f o u r g i v e n l i n e s . The p r i n c i p l e of c o n s e r v a t i o n of number can s t i l l be s a l v a g e d so l o n g as we  count  with m u l t i p l i c i t y  two.  At f i r s t  t h i s may  seem somewhat  c o n t r i v e d u n t i l we remember t h a t t h i s c o n f i g u r a t i o n o f l i n e s i s a c t u a l l y a degenerate case where, the two s o l u t i o n s of the g e n e r a l case have c o a l e s c e d i n t o one of the f o u r l i n e s , namely L^. become c l a r i f i e d when we  state i t l i k e this:  The p r i n c i p l e s t a r t s to  i f the number o f s o l u t i o n s  to an enumerative problem i s f i n i t e then t h a t number, counted w i t h m u l t i p l i c i t i e s i n the s p e c i a l case, i s the same as the number of s o l u t i o n s i n the g e n e r a l case. The problem of c o u n t i n g m u l t i p l i c i t i e s i t was  i s decidedly d i f f i c u l t ,  though  approached w i t h g r e a t courage, and m u l t i p l i c i t i e s were a s s i g n e d  w i t h g r e a t a l a c r i t y by the adept c l a s s i c a l geometers.  T h i s problem i s  c e n t r a l to the r i g o r o u s f o u n d a t i o n of Schubert's enumerative f a c t , i n t h e i r a r t i c l e GEometrie  Enumerative  of 1915,  calculus, i n  Zeuthen and  c o n s i d e r i t o f such fundamental s i g n i f i c a n c e t h a t they s t a t e t h a t i t s s o l u t i o n must have the h i g h e s t p r i o r i t y .  Pieri obtaining  - 6We w i l l r e t u r n t o d i s c u s s the problem o f m u l t i p l i c i t i e s f u r t h e r  after  a more c a r e f u l e x p o s i t i o n of the a l g e b r a i c and geometric i n t e r p r e t a t i o n s o f the p r i n c i p l e o f the c o n s e r v a t i o n o f number. G i v e n an enumerative problem, l e t us assume t h a t i t can be d e s c r i b e d by n homogeneous e q u a t i o n s i n n+1 homogeneous unknowns.  T h e o r e t i c a l l y we  can e l i m i n a t e v a r i a b l e s one by one u n t i l we o b t a i n a s i n g l e homogeneous e q u a t i o n i n two, homogeneous unknowns... The r o o t s o f t h i s e q u a t i o n c o r r e s p o n d to the s o l u t i o n s o f t h e o r i g i n a l system, thus the number of s o l u t i o n s , c o u n t i n g m u l t i p l i c i t i e s , o f the enumerative problem i s e q u a l to the degree of t h i s e q u a t i o n .  I t can be shown t h a t t h i s degree i s the p r o d u c t o f the  degrees o f t h e n e q u a t i o n s i n the o r i g i n a l system which a r e independent o f the c o e f f i c i e n t s .  Thus the. (weighted) number of s o l u t i o n s t o the enumerative  problem i s conserved under continuous v a r i a t i o n o f the parameters. However t h e r e are two snags.  Firstly,  t h i s argument i g n o r e s the  p o s s i b i l i t y o f extraneous r o o t s which c o u l d e a s i l y appear d u r i n g the e l i m i n a t i o n p r o c e d u r e , and s e c o n d l y , c o m p a r a t i v e l y few enumerative can be d e s c r i b e d i n such a simple way.  problems  So, a t the t u r n o f the c e n t u r y ,  Schubert's c a l c u l u s came under f i r e once a g a i n and, once a g a i n , i t s u r v i v e d . T h i s time G i a m b e l l i  (1904) and S e v e r i  (1912) r e s c u e d the c a l c u l u s i n t h e i r  papers b o t h c a l l e d S u l p r i n c i p i o d e l l a c o n s e r v a t i o n e d e l numero.  In these  papers G i a m b e l l i f o r m u l a t e d and S e v e r i developed the i d e a s t h a t put the Schubert c a l c u l u s on a geometric f o o t i n g . G e o m e t r i c a l l y , an enumerative problem concerns c o n d i t i o n s of i n t e r s e c t i o n or tangency on f i g u r e s o f a c e r t a i n type, and though we a r e only i n t e r e s t e d i n a f i n i t e number, of these, i t i s u s e f u l to l o o k a t the t o t a l i t y of a l l these f i g u r e s , f o r t h i s s e t can be i d e n t i f i e d w i t h a v a r i e t y .  We  say t h a t t h i s v a r i e t y p a r a m e t r i z e s the f i g u r e s i n q u e s t i o n , and we c a l l i t  - 7 the parameter v a r i e t y . algebraic  C o n d i t i o n s imposed on the f i g u r e s t u r n out t o be  ( i . e . d e f i n e d by p o l y n o m i a l e q u a t i o n s ) i n an enumerative  so the s e t of s o l u t i o n s to the problem which reduces  forms an a l g e b r a i c s e t .  problem,  A  condition  the freedom of the f i g u r e s by r parameters, i s c a l l e d an  r-fold  c o n d i t i o n and y i e l d s a subset of the parameter v a r i e t y of codimension  r.  Independent c o n d i t i o n s correspond to subsets i n g e n e r a l p o s i t i o n , sum  of  c o n d i t i o n s corresponds  to i n t e r s e c t i o n of s u b s e t s , product of c o n d i t i o n s  corresponds  to union of s u b s e t s , and e q u a l i t y of c o n d i t i o n s corresponds  what we  c a l l numerical equivalence.  now  S e v e r i , i n h i s p r e v i o u s l y mentioned a r t i c l e of 1912 S u i fondamenti  d e l l a geometria numerativa. e s u l l a t e o r i a  c a r a t . t e r i s t i c h e of 1916,  d e s c r i b e d the problem  and i n h i s a r t i c l e delle  g e o m e t r i c a l l y and  developed  an a l g e b r a i c i n t e r s e c t i o n t h e o r y , but t h i s o n l y s o l v e d the problem s e c t i o n s of h y p e r s u r f a c e s on the parameter v a r i e t y .  to  for inter-  Some i d e a s of P o i n c a r e  and Kronecker were developed by L e f s c h e t z (1924, 1926)  into a topological  i n t e r s e c t i o n theory u s i n g s i m p l i c e s , and van der Waerden r e c o g n i z e d t h a t t h i s theory was  s u f f i c i e n t l y g e n e r a l to g i v e the Schubert  f o u n d a t i o n , and d i d so i n 1930 K a l k U l s der abzahlenden  calculus a rigorous  w i t h h i s paper T o p o l o g i s c h e Begrundung  des  Geometrie.  A t o p o l o g i c a l i n t e r s e c t i o n theory f i r s t  r e q u i r e s the d i f f i c u l t  fact  that to each a l g e b r a i c subset can be a s s i g n e d a c l a s s i n the cohomology of the parameter v a r i e t y . are, h e u r i s t i c a l l y  Two  a l g e b r a i c subsets i n the same continuous f a m i l y  speaking, homotopic and consequently are a s s i g n e d the  same cohomology c l a s s .  The i n t e r s e c t i o n of two a l g e b r a i c subsets i n g e n e r a l  p o s i t i o n i s a s s i g n e d the cup product of t h e i r c o r r e s p o n d i n g cohomology c l a s s e s and t h e i r union the sum.  I t has a l s o been shown that i f a f i n i t e number of  a l g e b r a i c subsets i n g e n e r a l p o s i t i o n i n t e r s e c t i n a f i n i t e number of p o i n t s  - 8 then the degree of the product  of the c o r r e s p o n d i n g  cohomology c l a s s e s i s  equal to the number of p o i n t s i n the i n t e r s e c t i o n , and number does not change i f the a l g e b r a i c s u b s e t s , problem, are v a r i e d  consequently  this  i . e . the parameters of  continuously.  Though t h i s c o n s t i t u t e s a r i g o r o u s j u s t i f i c a t i o n of the p r i n c i p l e c o n s e r v a t i o n of number, inasmuch as we w i t h i n the c o n t e x t  of the c a l c u l u s of a l g e b r a i c cohomology c l a s s e s , we  c l a s s i c a l geometers have to be v e r i f i e d l i m i t s o f t h e i r v a l i d i t y " and  And  so we  the  "with ah exact d e t e r m i n a t i o n  i n such a way  still  For i n the statement of  the problem H i l b e r t makes i t c l e a r t h a t a l l the numbers, o b t a i n e d by  be  of  i n t e r p r e t the Schubert c a l c u l u s  cannot c o n s i d e r H i l b e r t ' s f i f t e e n t h problem s o l v e d .  s o l u t i o n s may  the  t h a t "the m u l t i p l i c i t y  of  of  their  foreseen."  r e t u r n to the problem of m u l t i p l i c i t i e s .  s t a t e d i n modern terms and,  T h i s problem has  i n theory, has been s o l v e d a b s t r a c t l y .  the parameter v a r i e t y , of the a l g e b r a i c subsets  d e f i n e d by  c o n d i t i o n s , at the p o i n t r e p r e s e n t i n g that s o l u t i o n .  to do t h i s e x p l i c i t l y .  on  the problem's  This d e f i n i t i o n  the d e s i r e d p r o p e r t i e s to s o l v e any m u l t i p l i c i t y problem but  difficult  been  The  m u l t i p l i c i t y of a s o l u t i o n i s d e f i n e d as the i n t e r s e c t i o n m u l t i p l i c i t y ,  all  the  has  i t is  T h i s however would not s a t i s f y  Hilbert  s i n c e he r e q u i r e s the e x p l i c i t n e s s and not j u s t a g e n e r a l method. What i s needed then i s a s e t of g e n e r a l p r i n c i p l e s that w i l l d e a l w i t h any m u l t i p l i c i t y without Classically, was  recourse  to. any  ad hoc methods i n a p a r t i c u l a r  i t seemed t h a t such a p r i n c i p l e was  t a c i t l y assumed, and  case.  this  t h a t i n the g e n e r a l case of an enumerative problem ( i . e . where the  f i g u r e s are i n g e n e r a l p o s i t i o n ) each f i g u r e s a t i s f y i n g the p r e s c r i b e d cond i t i o n s of c o n t a c t i s counted w i t h m u l t i p l i c i t y one.  T h i s seems to make  i n t u i t i v e sense, i n f a c t i t almost seems to be a t a u t o l o g y , but i n terms of the preceeding  i t s proof,  f o r m u l a t i o n of the n o t i o n of a m u l t i p l i c i t y , i s  -  by no means t r i v i a l .  The  in  an u n n a t u r a l way.  the p r i n c i p l e may Kleiman has unconditionally general  Kleiman has  unsolved.  the p r i n c i p a l i s v a l i d  t r a n s i t i v e l y on  ( i n zero c h a r a c t e r i s t i c ) f i g u r e s where  the  the parameter v a r i e t y .  even have complete knowledge of  complete t w i s t e d The  cubic  For  the v a r i e t y  space c u r v e s , f o r we  l a c k the  problem of a s s i g n i n g m u l t i p l i c i t i e s  as i s H i l b e r t ' s f i f t e e n t h problem i t s e l f ,  and  there  i s much to be  structure  i s deep, done  may.consider i t s o l v e d .  In view of the p r e c e e d i n g remarks our four given  t h a t t h i s example a r i s e s  o t h e r h i g h e r - o r d e r f i g u r e s however the problem remains  of i t s cohomology. r i n g .  b e f o r e we  out  trans-  i n any c h a r a c t e r i s t i c .  proved that  and  expected, i n  Thus the p o s s i b i l i t y remains t h a t some r e v i s e d form of  be v a l i d  do not  parametrizing  he p o i n t s  f o r l i n e a r spaces, moreover f o r any  cubics We  as might be  an example i n h i s paper The  t r a n s l a t e , but  l i n e a r group a c t s  quadries,  -  p r i n c i p l e does f a i l ,  positive characteristic. v e r s a l i t y of a g e n e r a l  9  example of the  l i n e s meeting  l i n e s i n 3 - s p a c e i s p a r t i c u l a r l y n i c e , the more so because i t i s  a l s o easy to v i s u a l i z e . i n t e r p r e t a t i o n s of the  We  now  exemplify the a l g e b r a i c and  Schubert c a l c u l u s i n t h i s  P r e s e r v e the n o t a t i o n  above and  l e t IK  be  any  geometric  way. f i x e d plane containing  L_^.  3  I f L meets any  l i n e a r space X,  a l i n e , or e q u i v a l e n t l y -1  in  IP  the  ( L n X) =0  dim  or 1 r e s p e c t i v e l y .  i s the dimension of the empty s e t , so,  skew l i n e s then  dim  ( L n L')  i n t e r s e c t i o n i s either a point  = -1.  The  (L n  >  By  or  convention,  i n p a r t i c u l a r , i f L and  c o n d i t i o n t h a t L meet L., l  L'  are  therefore  becomes dim  Now  L i s not  a priori,  dim  constrained (Ln IK)  to l i e i n any  > 0.  0.  p a r t i c u l a r plane containing  T h i s however i s not  K  and  an independent c o n d i t i o n  so, on  - 10 3  the s e t o f l i n e s  i n 3P .  The f a c t  that L i s contained  3  i n IP , though, i s  3 an independent c o n d i t i o n .  I t i s expressed as  dim (L n IP ) > 1 and means  3 implicitly  t h a t L i s not c o n s t r a i n e d  More g e n e r a l l y we can c o n s i d e r w i t h subspaces o f of  IP  n  i s called  ]P .  t o l i e i n a p r o p e r subspace of k - p l a n e s i n TP  The v a r i e t y p a r a m e t r i z i n g  n  n  and t h e i r i n t e r s e c t i o n s  k - d i m e n s i o n a l subspaces  the Grassmannian and i s denoted by G^( 3 P ) .  Any  n  imposed c l a s s i c a l l y  IP .  can be f o r m u l a t e d i n the f o l l o w i n g way.  condition  There i s a  s t r i c t l y - i n c r e a s i n g , n e s t e d sequence,  A  o  of l i n e a r subspaces of J P condition also  (Sch) and  C  n  A  l  C  ' " * \ C  1 , 1 1  such t h a t any. k-plane X s a t i s f y i n g  the imposed  satisfies  dim ( X n A . ) > i  vice versa.  5  0 < i < k  So a geometric c o n d i t i o n on a k-plane i n IP g i v e s n  to k+1 independent a l g e b r a i c c o n d i t i o n s .  I t c o u l d be shown t h a t  rise  the i - t h  c o n d i t i o n i s r.-"-fold where r ^ = ( n - k + i ) - dim ( A ^ ) , but i n p r a c t i c e a l l k+1 c o n d i t i o n s an r - f o l d  that  Instead  ( n - k + i ) - d i m . ( A . ) = (k+1) (n-k)  1=0 and  together.  we show l a t e r  that  (Sch) i s  c o n d i t i o n where  k V  r =  are considered  k V  -  [dim ( A . ) - i ]  1=0  1  1  (k+1)(n-k) i s the dimension of G ( J P ) . (Sch) i s c a l l e d n  c o n d i t i o n , and t h e s e t o f a l l k-planes s a t i s f y i n g  this  a Schubert v a r i e t y which we denote by n [ A Q , A ^ , . . . , A ^ J .  a Schubert  condition i s called It i s a variety  because i t s a t i s f i e s e x t r a l i n e a r e q u a t i o n s i n a d d i t i o n t o the q u a d r a t i c ones d e f i n i n g the Grassmannian which i s embedded i n p r o j e c t i v e space o f  'n+1 dimension  !k+l  - 1.  The Schubert v a r i e t i e s a r e then i n t e r s e c t i o n s o f the  - 11 Grassmannian w i t h c e r t a i n hyperplanes  N i n IP .  So the s e t o f l i n e s i n 3-space meeting  i s r e p r e s e n t e d by the  3 Schubert v a r i e t y ^ [ L ^ ,  IP ] and so the s e t . o f l i n e s meeting a l l f o u r g i v e n  l i n e s i s r e p r e s e n t e d by the v a r i e t y 3  4  i=l  3 Now t h e parameter v a r i e t y i n t h i s case i s G^( IP ) which has o n l y one d e f i n i n g q u a d r a t i c p o l y n o m i a l and thus i s a q u a d r i c h y p e r s u r f a c e i n Consequently  .  V i s d e f i n e d by one q u a d r a t i c , and f o u r l i n e a r e q u a t i o n s .  The  e l i m i n a t i o n i s o b v i o u s l y e a s i l y c a r r i e d out y i e l d i n g a s i n g l e homogeneous q u a d r a t i c p o l y n o m i a l i n two homogeneous unknowns and t h e r e f o r e the number of  l i n e s , i n g e n e r a l , meeting f o u r g i v e n l i n e s i n g e n e r a l p o s i t i o n i n  3-space i s e q u a l to the degree o f t h i s p o l y n o m i a l which i s 2 x 1 x 1 x 1 x 1 = 2. Our h i s t o r y so f a r has brought f o u n d a t i o n of the Schubert  us up t o 1930 and van d e r Waerden's  calculus.  With Ehresmann i n 1934 andi;his paper  Sur l a t o p o l o g i e de c e r t a i n s espaces homogenes the c a l c u l u s was put onto an even f i r m e r f o u n d a t i o n .  He showed t h a t the  2i  > Grassmannian w i t h c o e f f i c i e n t s i n TL i s generated s  Schubert v a r i e t i e s whose complex dimension groups a r e a l l t r i v i a l ) .  homology group o f the f r e e l y by the c l a s s e s o f  i s i , (the odd d i m e n s i o n a l  F o r t h i s reason a Schubert  variety i s also  \ r e f e r r e d to as a Schubert  cycle.  T h i s i s the f i r s t  p a r t of what i s c a l l e d  the b a s i s theorem and a t t h i s p o i n t we i n t e r r u p t the h i s t o r y . In  the f i r s t  two c h a p t e r s we d e s c r i b e i n d d e t a i l t h e parameter v a r i e t y  G^( IP ) and i t s Schubert n  subvarieties.  b a s i s theorem and show how the second along w i t h two formulae  We prove  (both p a r t s o f ) the  p a r t , which i s r e a l l y P o i n c a r e  due t o G i a m b e l l i and P i e r i p u t the Schubert  on a r i g o r o u s f o u n d a t i o n by a f f o r d i n g a complete d e s c r i p t i o n o f  duality, calculus  - 12 * n H (G iW ),7Z) as a 2 Z - a l g e b r a . In the l a s t  chapter we r e t u r n to our h i s t o r y , o u t l i n e some of the work  done s i n c e Ehresmann and d i s c u s s section theories  the l i m i t a t i o n s o f t h i s and o t h e r  t h a t have been developed s i n c e then.  m u l t i p l i c i t i e s a l s o o c c u r s i n the theory  inter-  The problem o f  o f s i n g u l a r i t i e s o f mappings which  we mention b r i e f l y , as w e l l as a d e s c r i p t i o n o f the s i n g u l a r l o c u s o f a Schubert v a r i e t y wherein we i n c l u d e some of our own o b s e r v a t i o n s  on the  matter. The  scope of H i l b e r t ' s f i f t e e n t h problem i s enormous.  mathematicians have c o n t r i b u t e d e f f o r t s have g i v e n b i r t h a l r e a d y born f r u i t . elusiveness,  A great  t o i t s p a r t i a l s o l u t i o n and t h e i r  many  collective  to new branches o f mathematics, many o f which have  But t h e r e remain those p a r t s  i n v i t e the c o n c e p t i o n  that, i n their  of even newer t h e o r i e s .  - 13 -  Chapter I.  THE §1  GRASSMANNIAN  The Naked Grassmannian In s t u d y i n g g e o m e t r i c a l o b j e c t s t h a t a r e "curved" one technique i s  to c o n s i d e r a l l the b e s t " s t r a i g h t " and  the q u e s t i o n a r i s e s  approximations  a s . t o where t o put a l l t h e s e .  c o n s i d e r t h e s e t o f a l l k - d i m e n s i o n a l subspaces o b j e c t , however, i s so i n t e r e s t i n g general foundation.  i . e . the tangent  dimension n, then the s e t o f subspaces  And so we a r e l e d to  of the ambient space.  i n i t s own r i g h t  F o r t h i s purpose,  spaces,  This  t h a t we g i v e i t a  l e t E ^ be a v e c t o r space of o f E o f dimension k i s c a l l e d the n  Grassmannian and denoted G, (E ) . k n Let  X e G, (E ) , then r e l a t i v e to some f i x e d b a s i s f o r E any k n n J  ordered b a s i s f o r X g i v e s r i s e t o a k* n m a t r i x over the ground which has rank k.  We c a l l  field  W  t h i s m a t r i x the S t i e f e l m a t r i x of the chosen  b a s i s , and the s e t o f a l l such S t i e f e l m a t r i c e s f o r a l l X e G, (E ) we k n the S t i e f e l space of k-frames i n E^, and denote i t by S t ( k , n ) . an a c t i o n o f the l i n e a r group GL(k, W) on St(k,n) by l e f t  call  There i s  multiplication.  S i n c e any o r b i t o f t h i s a c t i o n i s e x a c t l y t h e s e t o f S t i e f e l m a t r i c e s r e p r e s e n t i n g a l l t h e ordered bases o f a g i v e n subspace  i n G^(E_), the  Grassmannian appears as t h i s q u o t i e n t . Heuristically any o t h e r p o i n t .  s p e a k i n g , every p o i n t on the Grassmannian looks l i k e S t r i c t l y speaking t h e r e i s an a c t i o n of Aut(E^) on E ^  that induces an a c t i o n on the k-subspaces  o f E ^ which i s t r a n s i t i v e .  R e l a t i v e to a f i x e d b a s i s f o r E we have A u t ( E ) = GL(n, IF) and a n n decomposition E ^ = E' $ E" where E' i s the span of the f i r s t k b a s i s v e c t o r s and E" i s the span o f the remaining  (n-k).  I f two automorphisms  b o t h t a k e E' t o X e G. .(E ) then t h e i r r a t i o l e a v e s E' i n v a r i a n t , k n •  thus  G^CE^) can be r e l a b e l e d as the space o f l e f t c o s e t s o f t h e i s o t r o p y group o f E' i . e . GL(n, ] F ) / I s o t ( E ' ) . These two p o i n t s o f v i e w a r e r e c o n c i l e d as f o l l o w s :  a matrix  A e GL(n, IF) sends E' t o t h e subspace of E^ spanned by the f i r s t  k  columns o f A, so l e t us w r i t e A = ( A ^ l * ) and d e f i n e t h e map m: GL(n, TF) —  St(k,n) t  via  where A  fc  denotes t h e t r a n s p o s e o f A^.  T h i s map  i s clearly  surjective.  Consider the diagram  GL(n, IF)  c  m  *» S t ( k , n )  (1.1.1) G. (E ) k n  where ¥ i s t h e p r o j e c t i o n d e f i n e d above and <j> sends a S t i e f e l m a t r i x t o the span of i t s rows ( o r e q u i v a l e n t l y the o r b i t o f t h e GL(k, TF) a c t i o n ) . If  (A^|*) g GL(n, W)  then <f> o m( (A^| * ) ) i s t h e subspace o f E^ spanned by  t h e columns o f A^, whereas  ( ( A ^ J * ) ) i s the coset ( A ^ | * ) I s o t ( E ' ) which  c l e a r l y r e p r e s e n t s t h e same subspace, so we have t h a t (1.1.1) i s commutative. We n o t e i n p a s s i n g t h a t I s o t X E ' ) c o n s i s t s o f a l l m a t r i c e s o f t h e f o  N  rl-k  - 15 where 1^ e GL(k, I F ) , ^ _^ n  e GL(.n-k, IF), so Isot(.E') Is u s u a l l y w r i t t e n  GL(k,n-k, IF) . All  t h a t we have d e a l t w i t h so f a r i s the Grassmannian i n the  of " j u s t l i n e a r a l g e b r a , " and from t h i s , we  though we w i l l c o n t i n u e to get more mileage  are s p e c i f i c a l l y  d e r i v e d when  i n t e r e s t e d i n examining  has a g e o m e t r i c a l . f o u n d a t i o n .  then, we use the ambient spaces  H  and (E and  n  n  structures  the c o r r e s p o n d i n g G ( IR ) n  n  reason we  s t r u c t u r e t h a t we  that are  In d i f f e r e n t i a l geometry  and G, (<C ) have much i n common as d i f f e r e n t i a b l e m a n i f o l d s . k of the geometric  light  However most  examine i s p u r e l y a l g e b r a i c and f o r t h i s  a v o i d the h a s s l e s o f non. a l g e b r a i c a l l y c l o s e d f i e l d s .  Henceforth  then E i w i l l be the a f f i n e space o f dimension n over IF, which we  assume to  be a l g e b r a i c a l l y c l o s e d , and so i n the p a r t i c u l a r case where IF = (E we d e a l i n g with, two  t o p o l o g i e s on E^., the Z a r i s k i topology and  are  the u s u a l  topology. With e i t h e r topology on E^ = product of  IF  n  w i t h i t s e l f onto  IF  n  the p r o j e c t i o n from the n - f o l d  the f i r s t N  mapping.  GL(n, IF) i s t o p o l o g i c a l l y . IF  the determinant  k f a c t o r s i s a continuous open  2  - R, where R i s the zero s e t of  f u n c t i o n t h e r e f o r e c l o s e d : i n both t o p o l o g i e s .  On  the o t h e r  kn hand we  have St(k,n) i s t o p o l o g i c a l l y  m a t r i c e s w i t h rank s t r i c t l y  IF  -S,  where S i s the s e t o f  l e s s than k, but such a m a t r i x i s c h a r a c t e r i z e d  by a l l i t s k x k submatrices h a v i n g zero determinant, i n both t o p o l o g i e s . p r o j e c t i o n , and  so we  thus S i s a l s o  closed  C e r t a i n l y S i s c o n t a i n e d i n the image of R under the get a new  map  GL(k, IF)  which c o i n c i d e s e x a c t l y w i t h the map  >  St(k,n)  m i n diagram  are c l o s e d , m i s a l s o open and c o n t i n u o u s .  (1.1.1).  S i n c e R and S  By the p r e c e d i n g argument and  the  - 16 commutativity o f (1.1;1) we deduce t h a t Y and < J > induce i d e n t i c a l topologies  quotient  on G, (E ) . k n  We conclude t h i s s e c t i o n w i t h t h e o b s e r v a t i o n Grassmannian i s e x a c t l y showing that ^ ( E ^ )  t h a t when k = 1 the t o En ,  P ( E ) , the p r o j e c t i v e space a s s o c i a t e d  i s a g e n e r a l i z a t i o n o f one of the most important  concepts of geometry.  §2 T h e Grassmannian as V a r i e t y A great by  d e a l of the s t r u c t u r e present  s e e i n g how i t presents  itself  i n the Grassmannian i s a p p r e c i a t e d  as an a l g e b r a i c v a r i e t y .  As we have seen,  a s p e c i a l case of t h e Grassmannian i s p r o j e c t i v e space, so one would not be s u r p r i s e d to f i n d out that t h e Grassmannians a r e a l l p r o j e c t i v e v a r i e t i e s . To get an a l g e b r a i c h o l d on the p o i n t s o f G,(E ) the e x t e r i o r powers K. n k come t o hand e a s i l y . I f X i s a k-subspace of E then A X r e p r e s e n t s a n >  " l i n e through o r i g i n " i n t h e v e c t o r  for  space A E . Now, n k, ' k  l  a basis  k  k  X i f and only i f X^A...AX^ generates A A, so i f A X = A X' then X = X'  thus we get a c a n o n i c a l mapping. p : G, (E ) k n  +  !P(A E ) n k  which i s i n j e c t i v e . We show t h a t t h e image i s c l o s e d w i t h r e s p e c t on see  3P-(A^E ) ; and thus G, (E ) c l o t h e s i t s e l f n k n  t o the Z a r i s k i  as a p r o j e c t i v e v a r i e t y . J  e. A . i , 1  t h i s choose a b a s i s f o r E say {e,,...,e } then n 1 n  1 < i, < ... < i , < h k k represented  topology  . . A e .  l  To  :  ' k  k k i s a b a s i s f o r A E , so the p o i n t s o f TP (A E ) a r e n n  by t h e i r homogeneous c o o r d i n a t e s  (...,x. i  . ,,..) r e l a t i v e t o  1  . . . i  k  k n t h i s b a s i s , and an a f f i n e open cover o f IP (A E ) i s g i v e n by t h e (, ) s e t s n K.  17 U  . of p o i n t s w i t h homogeneous c o o r d i n a t e x. i ....-,.i 1 1  r  topology i t s u f f i c e s t o prove t h a t p(G  in  ,  ,  m  (E )) n U .  By  elementary  1  "k  .  i s Zariski  closed  U. L  l'  Without  l o s s of g e n e r a l i t y assume i . 3  E' be the span of e . . . , e  = j , and s e t U  „  = U.  Let  X , Z , . .., K  , E " the span of  1 >  x  . ^0. ,  k  \+i>•••>  e  a  n  d  x  t  h  e  n  s  P  a n  o  f  ^,...,x^, then each x_^ has a unique r e p r e s e n t a t i o n e^ + e\| where e^e E',  eVe E". x  So  X  ;  L  A . . . A x  k  + e^ACe^ +  =  =  e ' A e ' A . . . A e '  +  (terms t h a t a r e z e r o , or e l s e not  Thus p(.X) e U is  iff  =  e ^ A . . . A e ^  in f o r some  Xe^A.-.Ae^  a b a s i s f o r E' showing t h a t we  u. J -|_»"  „  " *' k  ,s  (1.2.1)  e^+w^,...,  1  k  =  e  +  where the w / s  1  A . . . A e / +  1  e^ + w^  where w.  to eE",  to c h o o s i n g the a f f i n e c o o r d i n a t e s  = x. . /x .We ^l'*""'^k 1>2,...,K  X,A...AX,  \ 4 0, but then e^,...,e£  c o u l d have chosen the e_^'s o r i g i n a l l y  g i v e X a unique b a s i s of the form e^ + w^, T h i s i s of course tantamount  + e^')  e^)A...A(.e^.  -.k  have  Y  i<i<  V e, A . •* , , , 1 l<x<j<k  A . . . AW. A.. . fcAe, 1  e1  k  . . AW. A . . . AW. A . . . Ae,  x  J  i  k  +  . . . + W,  1  A . . . AW.  k  a r e a l l i n the p l a c e denoted by t h e i r s u b s c r i p t , showing  t h a t p(G,(E )) n U  i s p a r a m e t r i z e d by the w.'s.  These w.'s  determine a k x ( n - k )  m a t r i x B r e l a t i v e to the b a s i s e. k+1  the  x^,....,x^ i s ( i l ' )  S t i e f e l m a t r i x of  k  B  =  B  i n turn w i l l e , n  i  s  i n fact  determined  - 18 completely from the second term i n (1.2,1) by  (1.2.2)  u  « l,z,...,i,...,k,j  = (.-l)  a... i j  k _ : L  1 < i < k k+1 < j < n,  where the c i r c u m f l e x over a s u b s c r i p t means that the s u b s c r i p t i s taken out. T h i s has the e f f e c t o f showing  t h a t p(G,(E )) i s covered by open s e t s each  c a n o n i c a l l y i s o m o r p h i c to a f f i n e space o f dimension k(n-k) and the r e s t o f the a f f i n e c o o r d i n a t e s a r e c l e a r l y r e l a t e d by p o l y n o m i a l s t o those of (1.2.2). find  Though some might be content to stop h e r e , we s h a l l p r e s s on t o  these r e l a t i o n s e x p l i c i t l y .  First  l e t us note t h a t the a l t e r n a t i n g  k k - l i n e a r form d e f i n e d on  X by the e q u a t i o n i=l  P. l» J  , , , , : ,  k  (X) = P. l'' 3  (x , -  ...,x ) = u.  ' k  2  J  V '  ,  k  2  i s independent o f t h e c o e f f i c i e n t s o f a l l the e.'s, save e. ,...,e. , i n  the expansion o f the x_/s and so i t i s a c t u a l l y a f u n c t i o n o f the rows o f the k x k submatrix o f the S t i e f e l m a t r i x formed by t a k i n g the J - ^ * • • • » j j ^ columns.  As such P. l' J  ,  ,  ,  ,  J  . k  1  must be a non-zero, s c a l a r m u l t i p l e o f the  determinant f u n c t i o n ; but t h i s s c a l a r i s c l e a r l y independent o f the rows chosen s i n c e P., l"*J  , : i  ., w i l l perform, e x a c t l y the same sequence o f a r i t h m e t i c k  o p e r a t i o n s on columns j ' ,.. . j ' to o b t a i n x., 1 k i t s columns t o o b t a i n x. . . J -.-»J 1 >  it  ,, as P. . l'"""'~'k  Denoting t h i s k x k m a t r i x by A. 2  k  f o l l o w s from t h e o b s e r v a t i o n o f P^ ^  does on  3  ^(x^,...,x^) = 1 t h a t  . , V'-> k 3  :  - 19 -  (1.2.3)  u. 1' 2  . = det(A. .) 2' * * *' k ~'l'*'"'~'k  2  2  Expanding by minors along the  (1.2.4)  u. 2  .  .  = J  l' 2'"' k  ct i where A.'  ("D  a + 1  1=1  2  2  row gives us  a . detCA?' a  i  J  2  .)  1  V "  ,  2  k  . denotes the (k-1) x (k-1) matrix obtained from A.  J-^» • • • »  J  -  by d e l e t i n g the a^ row and i " * 1  1  1  ^  column.  i s a l l zeroes except f o r a 1 i n the r e p l a c i n g the i  ^ column of A.  detiA®'  1  V "  3  k  1^  = 1L  The a*"* column of A 1  . b y the  . ) = det(A  •••>  row, so c l e a r l y we have, on  jj>...»J  (1.2.5)  >  column of A  „  1  , ,  X,Z,...,K  k  .- :  a , i  ll"-" !^, 2  A'  where the circumflex beneath the subscript means that i t replaces the one taken out.  Furthermore replacing the a*"* column of A. „ , by the i i j Zj • • • j K column of A. . w e obtain, as i n equation (1.2.2) 1  J  l  5  - - . . J  (.1.2.6)  k  a'.. = det(A . ) a. 2 ^ J-»...»o»J^>...»k A  and so on combining (1.2.3), (1.2.4), (1.2.5), (1.2.6) we conclude that k u.  . -  3j> "**' k 3  i=l  (-1)  u..  .  , u.  J->...>ct,J^j...»K-  or i n homogeneous coordinates  A  .  j^,...,J^,^,-...jj^  =  0  ^  -  20  (1.2.7)  x  ,x.  . - Y  X , . . . , K. J i . ' ' " * ' ~ ' k  i=l  where i t i s understood  (-1)  06  4- " X  x  „  .  x,. .. j 0 t j j ^ j . . . j l £  t h a t the x.  .  x.  A  .  J 2> • • ' » J  >• » • ' J j ^  = 0  are a l t e r n a t i n g i n t h e i r  indices.  Thus we have shown t h a t f o r any Xe p(G, (E )) n U t h e r e i s a p o i n t i n IP (A E ) k n n whose homogeneous c o o r d i n a t e s s a t i s f y 0 < J  < J  1  2  < • • • < J  C o n v e r s e l y we  (1.2.7) f o r 1 < a < k,  < n.  k  show t h a t any p o i n t  (...,x. . , ...) x^ > • • • > ^  e  IP(A E )  k  ^ t  s a t i s f y i n g x^ p(G  (E )) n U. k n  u  a  Without  n  e q u a t i o n s (1.2.7) i s indeed a p o i n t i n  d  l o s s of g e n e r a l i t y we may  Define a kx n matrix ( ^ j ) ( a  ii  a  i-  a s  =  X  But i f 1 < i < k then a.. =6.., ij ij  equation  n  l  x  assume t h a t x  , = 1.  X , • • • , K.  (1.2.2))  k i '  the Kronecker d e l t a , thus '  (a..) i s the xj  S t i e f e l Matrix (  |B) of a k - d i m e n s i o n a l subspace X of E  such  p  Now  by  n  „  u(^)  =  !•  c o n s i d e r the m a t r i x formed  from  that  replacing  one of i t s columns by a column from B, then the determinant of t h i s m a t r i x , k-i i . e . p. <» , . (x) i s simply (-1) a.., but s i n c e the r e s t of the X,...,X,...,K,J  c o o r d i n a t e s p. we  XJ  . (X) are generated by these a c c o r d i n g to (1.2.7) then  have  P. V "  f o r a l l sequences We  . (X) = x. ,  3  k  J  l'**"' k J  j^,...,j « k  summarize the above d i s c u s s i o n i n the f o l l o w i n g theorem:  - 21 Theorem (.1.2.8) The mapping p : G, (E ) K  >-IP.(.A E ) i s a c l o s e d embedding, g i v i n g  n  XI  G ^ ( E ) the s t r u c t u r e of a n o n - s i n g u l a r p r o j e c t i v e v a r i e t y of  dimension  n  k(n-k).  The  P.  . (X) are c a l l e d l'"* .k p i s c a l l e d the P l u c k e r embedding. J  the P l u c k e r . c o o r d i n a t e s of X  T h i s theorem allows us to make p r e c i s e the n o t i o n of Grassmannian as a parameter v a r i e t y . k-planes  When we  between the s e t of k - d i m e n s i o n a l v a r i e t y p(G, (E ) ) . k n  Henceforth  the  say t h a t G^(E  i n n-space we mean t h a t t h e r e i s a one-to-one  G, (E ) and  and  , J  subspaces of E^ and  )  parametrizes  correspondence  the p r o j e c t i v e  there i s no need to d i s t i n g u i s h between  i t s image under the P l u c k e r embedding and  so.we i d e n t i f y  the  two. A p o i n t X £ G^(E^) i s the s o l u t i o n space to a system of homogeneous l i n e a r equations w i t h rank ( n - k ) .  S i n c e t h i s system e q u a l l y w e l l d e s c r i b e s  c o n d i t i o n s on the homogeneous c o o r d i n a t e s of l i n e a r subspace  1P(X), G (E ) may ic n  ^"(E ) y i e l d i n g n  a l s o be thought  the p r o j e c t i v e  of as p a r a m e t r i z i n g  the  ( k - 1 ) - d i m e n s i o n a l p r o j e c t i v e l i n e a r subspaces of p r o j e c t i v e ( n - l ) - s p a c e . We  write  W The  B  G  k - i  (  p  (  E  ) f  )  B  i  G  k - i  {  F  n  '  1  (  )  )  -  s i m p l e s t example of a Grassmannian which i s not a p r o j e c t i v e space  i s G^CE^) .  In the case where  IF =  (E, t h i s i s the same as the space  3  G,( IP ) which i s mentioned i n the i n t r o d u c t i o n . 1 and  I  so i t i s a q u a d r i c h y p e r s u r f a c e i n  3P^( IF) h a v i n g  equation  X  12 34 X  X  13 24 X  +  X  G„(E.) has dimension 2 4  1 4 2 3 " °' X  4  the s i n g l e d e f i n i n g  - 22 In the next s e c t i o n we view t h e Grassmannian as a complex m a n i f o l d . If  IF =  IR, G, (E ) i s . a r e a l m a n i f o l d a l s o , but we do not d i s c u s s t h i s f o r k n  reasons, mentioned b e f o r e .  I n e i t h e r case, however, the Grassmannian i s  compact, b e i n g a c l o s e d subset of p r o j e c t i v e space which i s compact.  §3 The Grassmannian as M a n i f o l d The r e s u l t s o f S e c t i o n s 1 and 2 can be a p p l i e d immediately the s t r u c t u r e of G (E ) as a complex m a n i f o l d . K.  to study  In the course o f p r o v i n g  XI  Theorem (1.2.8) we e s t a b l i s h t h a t G, (E ) i s covered by open s e t s k n J  W. = G (E ) n U. . J^>"'-»Jk k n -'l'*""'^k  which a r e a l l c a n o n i c a l l y isomorphic as  a f f i n e spaces of dimension k ( n - k ) .  These isomorphisms, i n the case where  ]F = (C, a r e a l s o b i h o l o m o r p h i c and so G, (E ) i s the complex m a n i f o l d a s s o c i a t e d to the a l g e b r a i c v a r i e t y o f S e c t i o n 2. We  can see these c h a r t s a r r i v i n g i n a s l i g h t l y d i f f e r e n t way  from  kn S t ( k , n ) , which, now b e i n g an open subset o f (C  w i t h the u s u a l t o p o l o g y ,  takes i t s r i g h t f u l p l a c e among the m a n i f o l d s . S t i e f e l m a t r i x A l e t A.  .  J 2»• • •»J  the s e t V. . l ' •' • k J  As i n S e c t i o n 2, f o r any  be the m a t r i x of columns . j .  then k  k  = {AeSt(k,n);  , J  j  . W O }  det(A. J  l '  ,  ,  ,  ,  J  k  i s a Z a r i s k i open subset of the S t i e f e l m a n i f o l d which i s e v i d e n t l y under t h e a c t i o n of GL(k,(C) and so the image cp CV. l J  ,  ,  -  ,  ,  J  stable  . ) i s an open k  s e t i n G (E ) which i s , of course, the s e t W. . above, k n 1' * * ' '^k 3  As we have remarked i n S e c t i o n 1, the Grassmannian i s the same a l l over. The group GL(n,(C) a c t s t r a n s i t i v e l y by automorphisms which a r e l i n e a r , whence a l g e b r a i c and holomorphic,  and so G^(E^) earns  the t i t l e  of a  homogeneous space. way when we  T h i s f a c t can a l s o be seen i n a s l i g h t l y  different  t h i n k of (C as endowed w i t h i t s u s u a l h e r m i t i a n i n n e r p r o d u c t . n  The t r a n s i t i v e a c t i o n i s now g i v e n by the u n i t a r y group U ( n ) , and the i s o t r o p y o f E' i s denoted U ( k , n ^ k ) .  Consider f i r s t  t : GL(n,(C)  the continuous  map  > GL(n,C)  via A  > AA  A  where A  denotes the conjugate t r a n s p o s e of A.  image under  t  closed i t s e l f .  o f the c l o s e d s e t c o n s i s t i n g of the i d e n t i t y t^, From the e q u a t i o n AA n  z . . = z. .z .  .^  = H n  V  of  and so i s  f o r A = (z..) eU(n) we have, ij  z. .z . . =  showing t h a t U(n) i s both c l o s e d and bounded, can  U(n) i s the i n v e r s e  1  i . e . compact.  S i n c e G, (E ) k n  be i d e n t i f i e d as U(n)/U(k,n-k) t h i s p o i n t o f o f view has the advantage showing t h a t the Grassmannian i s compact, w i t h o u t v e n t u r i n g i n t o the  algebraic category.  We would, however, have ended up n a t u r a l l y a t  p r o j e c t i v e space anyway, s i n c e i n t e r e s t i n g compact, complex m a n i f o l d s  can't  l i v e i n an a f f i n e environment. I t would be unwise to c o n t i n u e to s e p a r a t e the d i s c u s s i o n  into  d i s t i n c t c a t e g o r i e s s i n c e p a r t of the charm of the Grassmannian i s how v a r i o u s s t r u c t u r e s f l o w i n t o each o t h e r .  the  H e n c e f o r t h then we s h a l l assume  t a c i t l y a l l the s t r u c t u r e r e q u i r e d by the c o n t e x t .  - 24 §4  The  U n i v e r s a l Bundle over the Grassmannian  Considering  the Grassmannian as a complex m a n i f o l d  ( i . e . E^ = d )  we  n  can d e f i n e a bundle over G, (E ) of rank k which has some u s e f u l p r o p e r t i e s . k n To each p o i n t X e G, (E ) we must a s s o c i a t e a k d i m e n s i o n a l v e c t o r space and k n the space X i t s e l f  i s a natural choice.  e x i s t e n c e of l o c a l t r i v i a l i z a t i o n s and  We must now  demonstrate the  show the c o m p a t a b i l i t y of these  on  the i n t e r s e c t i o n s by e x p l i c i t l y d e f i n i n g the t r a n s i t i o n f u n c t i o n s . To t h i s end  r e c a l l the map  0 : St(k,n) -> G (E ) as d e f i n e d i n s e c t i o n 1 K.  and  the open cover {W^.}  n  as d e f i n e d i n s e c t i o n 3 where I = ( i , . . . , i  i s easy to see t h a t 3> i s holomorphic i n t h i s case.  Denote by U  ).  It  the s e t K.  c o n s i s t i n g of a l l p a i r s . ( X , x ) where X e G. (E ) and x e X, so k n U, c G, (E -) x C . Define k k n  that  n  JJ : U, k  G. (E ) k n  via (X,x) ->  X  -1 k and we must f i r s t e x h i b i t homeomorphisms F^. : IT (W^.) W^x (£ i . e . we k -1 to use (E as a c a n o n i c a l model of each of the f i b r e s TT (X) f o r each }  XeWj.  Now,  f o r a given X e W  t h a t the k x k  property  a S t i e f e l m a t r i x A a s s o c i a t e d to X has  submatrix A^  columns i s n o n - s i n g u l a r ,  and we  formed by  can without  t a k i n g the i - j ^ '  : (X,x) On  <E  -> (X>v)  r e l a t i v e to the c a n o n i c a l b a s i s . i s then the r e q u i r e d  the o v e r l a p Wj. n Wj,  and such t h a t A  t a k i n g columns j ^ , . . . , J k  ^2^'  '^k^  The  the  3J^.  coordinates  mapping  one.  X i s represented  i s n o n - s i n g u l a r , where A  the  l o s s of g e n e r a l i t y assume A^. =  A v e c t o r x e X i s a l i n e a r combination of the rows of A, and of a v e c t o r V e  wish  by a m a t r i x A such t h a t A^. =  i s the k x k  ll^  submatrix formed by  The m a t r i x A, r e p r e s e n t i n g X and h a v i n g  A^. = ll ^ i s  - 25 unique, l i k e w i s e the m a t r i x A-  r e p r e s e n t i n g X and h a v i n g A^ =. i  A = T A' where T = A e GL(_k, X X J  C)  F F" J- J i s g i v e n by  (X,x) — •  n W  : W JL  J  x c  -> W  k  n W x <c J_ J  k  IJ  :  w  T  n  W  J  G L ( k  ' ) c  i s e v i d e n t l y holomorphic.  A.  The homeomorphism  (X,T (x)) and the mapping  t  g i v e n by X —>• T  1  i s unique.  . thus  Thus U  K  i s a holomorphic v e c t o r  bundle of rank k over G, (E ) . k n We  can d e f i n e n g l o b a l s e c t i o n s , S , of U, over G, (E ) as f o l l o w s : l e t a k k n S : W a, I I T  be d e f i n e d by S column o f A.  W  T  x <E  k  I  ( X ) = (X,C ), where A i s as b e f o r e and C  a  a, iT  T  S i s c l e a r l y holomorphic and S. a, I 1^,1  f i b r e over W^..  I t remains to show that t h i s way  i s the  a  ,...,S. """k  generate each  of d e f i n i n g a s e c t i o n i s  t r u l y g l o b a l , i . e . that i t i s compatible w i t h the t r a n s i t i o n f u n c t i o n s on overlaps. of A',  If XeW  T  J  a l s o , then S  and A' i s as b e f o r e .  ( X ) = (X,C') where C' i s the a "* a,J a 1  T  But we  C  a  = A  a  see immediately  1  column  that  C' J a  which i s a l l t h a t i s r e q u i r e d f o r the p a t c h i n g . The f o l l o w i n g theorem j u s t i f i e s  the usage of the a d j e c t i v e  "universal"  when r e f e r r i n g t o U^. Theorem: L e t M be a.complex m a n i f o l d of dimension n.  I f K->  M i s a holomorphic  v e c t o r bundle rank k, generated by n g l o b a l s e c t i o n s r^,-. i .-,-r , then t h e r e i s a holomorphic  map  <j> T  26  -  M -> G. (E ) k n  such t h a t K i s the induced bundle A T  (U, ) and r . = A ( S . ) . k 1 1  This r e s u l t i s  i n c l u d e d f o r the sake of completeness and i s r e f e r r e d to o n l y b r i e f l y ,  so we  r e f r a i n from r e p r o d u c i n g the p r o o f h e r e . The u n i v e r s a l bundle U, i s a subbundle of the t r i v i a l bundle G, (E ) x ( C n . k k n We the  denote the q u o t i e n t bundle by Q  u n i v e r s a l q u o t i e n t bundle on G, (E ) . 0 -»• U, k  of  §5  ^ which has rank n-k and which i s c a l l e d  G, (E ) k n  i£  x  C  Thus the sequence  XTn  Q  , -* 0  n-k  bundles over G, (E ) i s e x a c t , k n The Dual I f we  Grassmannian  c o n s i d e r f o r a moment a k - d i m e n s i o n a l l i n e a r subspace X o f  3R  n  we see that there i s a unique ( n - k ) - d i m e n s i o n a l subspace c o r r e s p o n d i n g to X  i . e . the o r t h o g o n a l complement X^ r e l a t i v e to the u s u a l orthonormal An isomorphism G, ( IR ) ^ G , ( 3R ) k n-k  basis.  not  n  work f o r an a r b i t r a r y ground f i e l d  n  i s then o b v i o u s , but t h i s does  and depends on a c h o i c e of b a s i s .  The i d e a that the s e t of k-planes i n n-space'should l o o k l i k e the set  of- (n-k)r-planes i n n-space can be f o r m u l a t e d n a t u r a l l y  let  E  n  as f o l l o w s :  = Horn ^ ( E , IF), where E i s a v e c t o r space of dimension n over an JF n • n  a r b i t r a r y ground f i e l d •  IF.  D e f i n e , f o r X e G . (E ) , k n  : f (x) = 0, x e X} . n (X°) = n-k, so that we have a map  X ° = {f e E One  checks t h a t dim  It  d : G ,(E ) -*• G (E ) k n n-k n  which i s e a s i l y seen to be a s e t isomorphism. the  map  The i n v e r s e can be g i v e n by  - 27  G  -  , ( £ } - • G (I ) n-k n k n  = { A e E : d>(f) = 0 o n  as defined by Y —• Y , where Y o the canonical isomorphism.  G, CE ) k n f e Y } and where n i s  A  G  , (E ) i s c a l l e d the dual Grassmannian. n-k n In the case where IF = <C, d i s a complex a n a l y t i c isomorphism.  Consider the u n i v e r s a l bundle U  , over G , (E ), the induced bundle n-k n-k n d (U , ) i s the dual of the u n i v e r s a l quotient bundle on G, (E ). We have n-k k n the exact sequence of bundles on G  o+.u  ,  -»•  n-k _1  and Q • = (d k  *  ~  G  . (E )  n-k ~  , (E ), n—k n n  x  e  n  ^  q,  4  o  k  ) (U, ) where U, i s the dual bundle of U, . k k k  - 28 -  Chapter I I  THE §1  The  Definition  We have up to now its  SCHUBERT VARIETIES  c o n s i d e r e d the Grassmannian as a completed form, but  true f a s c i n a t i o n l i e s i n s i d e .  The alignment o f the kr-dimensional sub-  spaces w i t h each other p r o v i d e s a means o f c l a s s i f y i n g one was  them even though each  p r e v i o u s l y u n d i s t i n g u i s h e d by v i r t u e o f homogeneity.  To examine  t h i s alignment we  the chosen b a s i s e, 1  c o n s i d e r the f i l t r a t i o n on E^ determined by  e , that i s n 0 = E c E , c . . . c E  (2.1.1)  o  1  where E . i s the span of e,-, ...,e.. l 1 x  We  n can t h i n k of E , as h a v i n g the "best k °  a l i g n m e n t " w i t h t h i s f i l t r a t i o n and compare the o t h e r p o i n t s X of G, ( E ) to k n E ^ by comparing the s i z e s of E ^ n E_^ and X n E_. . To t h i s end then we c o n s i d e r n \ which we c a l l the i n t e r s e c t i o n 1=0 the sequence of i n t e g e r s j^dim (X n sequence of X and denote by i(.X) .  i ( E ) =. and t h i s i s our b a s i c  In p a r t i c u l a r we  have  (0,1,2,3,...,k,k,k,...,k)  sequence.  i ( X ) i s always a non d e c r e a s i n g sequence s t a r t i n g a t zero and eventually  constant w i t h v a l u e k.  expansion, so i n t u i t i v e l y we most one.  A t each s t a g e we a l l o w one more dimension  should expect jumps i n the sequence of h e i g h t a t  T h i s i s seen to be t r u e by i n s p e c t i n g the p a i r of exact i+1  0 — •  X n E. — •  X n E . ,,  -x 4 - —T+ JF  0 —>  X n E  x nE  — X n E _ / X n E . —> X+l X  i  — •  X  becoming  x+1  X+X  0  sequences  - 29 where x^"*" i s the p r o j e c t i o n onto the ( i + l )  X n E . , , / X n E . has dimension 1 or 0 depending l+l l st any  (i+1)  coordinate.  on whether or not X n E . , _ has 1+1  A g e n e r a l i n t e r s e c t i o n sequence  C-0 J 0 J • • * j O y X j - X y a • • j X « 2 • 2 ^ • • • y 2 j • • • y  where the z e r o t h p l a c e i s always  the  that i s  T h i s argument shows t h a t t h e r e a r e e x a c t l y k p l a c e s  where the dimension jumps.  Comparing  coordinate;  s t  X y lt 1  then, l o o k s  like  y a • • y lC~* X j lC j lC j a a a y k.)  —  zero.  i ( X ) t o i ( E ) we. see t h a t . t h e d i f f e r e n c e i s i n the p l a c e where  dimension jumps f o r the i ^ .  time.  For i ( E ^ )  the 1^  jump o c c u r s a t the  th i  p l a c e , but i n g e n e r a l t h e r e i s a l a g of say, a^.  dim(X n E ) =. i but dim a .+i i and,  ( X n E, a  l  , - -•) = i - 1 . +1—1  i n t u r n , a r e u n i q u e l y determined by  In o t h e r words  These l a g s u n i q u e l y determine  i(X).  A p r o p e r t y of i ( X ) i s t h a t each i n t e g e r 1 < i < k appears a t l e a s t but  the number of times i t does appear i s e x a c t l y  thus ^+i~ a  of  a  ±  ~ ®'  Thus we have a b i j e c t i v e  i n t e r s e c t i o n sequences  ( a ^ ^ + i+'l) +  and the s e t JJ, of sequences We  i n c r e a s i n g sequences  of  how  But we to  JJ and  1 < a | < a^ < ••• ^ a ^ < n, so we r \ n number of d i s t i n c t i n t e r s e c t i o n sequences i s We  can t h i n k of the sequence  (a  awkwardly the k-plane X s i t s can also, use  and  (a ,...,a ) such  that  K.  note h e r e t h a t the s e t mapping (a ,...,a )  (a^+1,a2+2,...,a^+k) g i v e s a b i j e c t i o n between  the  + i)  correspondence between the s e t  J_  0 < a^ < ... < a^ < n-k.  - (a  once,  JK, the s e t o f  —>  strictly  see immediately  that  a ) as a measurement, i n some sense,  relative  to the chosen f i l t r a t i o n  on E.  (a^,.,.ya^) as a bound on f a r we a l l o w t h i s awkwardness  range as we v a r y the k-plane X ,  So we  c o n s i d e r the s e t o f k-planes whose th  i n t e r s e c t i o n sequence has a .lag o f a t most a^ i n the p o s i t i o n of the i Let  us denote t h i s s e t fi('a^, . . ,, a^) .  E q u i v a l e n t l y , but more c o n c i s e l y  jump.  - 30 ft (a, , I  ,£1, ) - {Xg G, (E ) ; dim (XnE , .) > i , 1 <; i<; k . k k n • +i — • —' — 1 " This set can be described by r e l a t i o n s among the Plucker coordinates x. X  a  . b y the f o l l o w i n g : l'•••' k 1  P r o p o s i t i o n (2.1.2) fi(a^,...,a )  i s the subvariety of ^^(E^) corresponding to the l i n e a r  k  polynomials x. . where j , . .., j 1' " " k  i s any sequence such that j > a .  +X A  f o r some 1 <_ A <^ k. Proof: Let Xefl(a^,...,a^) and l e t j > a^ >-'X f o r some 1 <_ \ <_ k. dim(Xn E _^) >_ i , 1 <_ i <_ k we may. choose a b a s i s , x^, . . . i a +  such that  x.e E , ., so the S t i e f e l matrix of x, ,...,x, looks l i k e x a.+x' 1' k x  X  1 1 1 2 ••' l , + 1 X  X  a i  X  X  21 22 X  x  00 x  2  A1 X2 ••• A,a£fl X  0  2 , a + l ••• 2£ +2  *X^+2  X  x,kl^k2 „x '•• k , X  consequently P.  a ; L  +l  X  k'a +2 2  00  0  +\  "k.a^+X  00  "k^+k  .. (X) i s the determinant of the matrix  Since  00 ... 0  31  -  1  o  x...  '\*-l  X  X  x.  A+I,j  *  1  *'  .  x  ^•>3  X  1  Using  ^A-I  XA+1  k  '  J  A - l  .  k,3  .  . (X) = 0,  thus  X  .  k,J  the L a p l a c e expansion of the determinant  P.  .  x  the l i n e a r p o l y n o m i a l s  we  k  get d i r e c t l y  that  a l l v a n i s h on fi(a ,...,a  )  Conversely,  c o n s i d e r a p o i n t X of G, (E ) whose P l u c k e r c o o r d i n a t e s k n s a t i s f y the l i n e a r r e l a t i o n s . We pass to the a f f i n e c o o r d i n a t e s on W  where  i s the sequence, chosen from among those f o r which k x ^ 0, which maximizes the sum J. j . From §2, Chapter I we know J'V"' i > • k• • »Ji = l ~r -£•1  9***9  ...£  <£i_  J  r  t h a t the p o i n t w i t h P l u c k e r c o o r d i n a t e s  whose S t i e f e l m a t r i x  u,,  the m a x i m a l i t y  of  Now  we  f o r any  k  k  r=l  r=l r^i  Y., £ r=l  we  r  In the same way  has  a b a s i s x^,...,x  .k-i i s (x. .) = ((-1) u„  p t 0 then £. < a.+i. 1'"' * ' v i —  so by  x.  get f o r i ' < i  ).  j >  get x. . = 0 . 1 3  a . + i we  Since  have  T h i s shows t h a t x 1  sE i a  +  :  - 32 -  l thus dim ( X E n  & +  i  ^ ) > i p u t t i n g X i n f2(a  .,.  > a  )  q.e.d.  i Note t h a t fi (a, > • • • »'a. ) e. G (E ) corresponds t o the v a r i e t y  Q[A  o  , ...,A. , ] C G , , (1P(E ) ) as d e f i n e d Tc-1 - k-1 n  i n the i n t r o d u c t i o n where A. = P ( E ,.) I a.+x l  and so we a r e c o m p l e t e l y j u s t i f i e d i n c a l l i n g ft(a ,...,a^) a Schubert v a r i e t y . The c o n d i t i o n dim (Xn E, , .) > i a.+i x  1 < i < k  i s c a l l e d a Schubert c o n d i t i o n , which i s a l s o c o n s i s t e n t w i t h the d e f i n i t i o n i n the introduction.  F i n a l l y , f o r convenience, we c a l l  ( a ,...,a ) a Schubert K.  X  symbol over G, (E ) . k n Some examples a r e i n o r d e r h e r e . (2.1.3)  n(0,0,...,0)  In t h i s case the k-planes a r e not allowed to roam a t a l l , v a r i e t y must c o n s i s t o f j u s t E^.  To.see t h i s p r o p e r l y  dim ( X n E ^ ) > k f o r any X g f t ( 0 , . . . , 0 ) . both X and E  and so t h i s  note t h a t , i n p a r t i c u l a r ,  T h i s must be an e q u a l i t y however,  since  have dimension k b u t then we must have X = E .  (Z.1.4) ft(0,0, ... ,0,a,a,. • . ,a) where the number of a's i s d, 1 < d < k For X fi(0,... ,0, a, • • •, a) a g a i n i n p a r t i c u l a r dim (X n E ^ ^ ) ^ k-d and so e  c.X,  S i m i l a r l y s i n c e dim (X n E ^ )  C o n v e r s e l y suppose E, , c g c E . k-d a+k J  dim (X nE.) x  > k we have  x E^ c  + k  .  F o r 1 ^ i ^ k-d E. X x C  thus  > i , and f o r k-d+1 ^ i ^ k we have  dim (X n E , .) = k+cc+i - dim (X + E , .) > i ct+x a+i s i n c e X + E ^ ^ c o+k' E  S  °  £2(0,. .., 0, a,.. ., a) , thus  X  satisfies  the requirements f o r l y i n g i n  - 33 n(P,...,0,a,...,a) = { X  e  GkCEn) ; E ^ c X c  E^}  and by sending X t o X / E _ ^ we see t h a t £2 (0, . . , ,0,a, , . • ,a). i s isomorphic to k  V W  V d =V V d ) • }  C2.1-.5). £2(n-k,n-k,. .. ,n-k) T h i s v a r i e t y a l l o w s the l a r g e s t p o s s i b l e l a g s , consequently Schubert v a r i e t y .  dim  the l a r g e s t  I t i s i n f a c t the whole of G, (E ), s i n c e f o r any X e G, (E ) k n k n  (Xn E . , .) = dim (X) + dim (E , , .) - dim (X + E , , .) " n-k+i n-k+i n-k+x = n+i - dim (X  E , ,.) n-k+x  but c e r t a i n l y dim (X + E , , .) < n because X+E. . c E, t h e r e f o r e n-k+x n-k+x~ J  dim  §2.  (X  E  , .) > i and Xe£2(n-k,n-k,...,n-k).  n—K+X  Example: The Schubert V a r i e t i e s i n G ( E ^ ) 2  We now i l l u s t r a t e the p r e v i o u s s e c t i o n by attempting s u b v a r i e t i e s of ^2^r?  geometry of the Schubert  '  t o v i s u a l i z e the  T h i s Grassmannian, as we  have mentioned b e f o r e i s a q u a d r i c h y p e r s u r f a c e i n TP'* w i t h the s i n g l e defining  equation  X  12 34" 13 24 X  X  X  + X  14 23 X  =  °-  The Schubert v a r i e t i e s a r e 0,(0,0), £2(0,1), £2(0,2), £2(1,1), £2(1,2) and £2(2,2). From example  (2.1.3) we have that  £2(0,0) =  from example (;2.1,4) we have t h a t  - 34 52(0,1) = G C E ) =  JP ,  9.(0,2) £ G ( K ) s  JP ,  1  3  fiU.l) and  from example  £  G  (  1  2  2  3  E  2  3  )  = 1 G  ( E  3  )  =  (2.1.5) we have t h a t  £2(2,2) = G ( E ) . 2  4  The remaining Schubert v a r i e t y £2(1,2),. i s more d i f f i c u l t the  f a c t that i t i s not smooth.  to d e s c r i b e due t o  I t i s the s m a l l e s t example of a s i n g u l a r  Schubert v a r i e t y , and the remainder of t h i s s e c t i o n i s devoted t o i t s description. In  a d d i t i o n to the q u a d r a t i c r e l a t i o n above, (2.1.2) t e l l s  $2(1,2), we have the r e l a t i o n x^^-=  Note t h a t  1 2  x  3 4  - x  1 3  x  2 4  = V(x  1 4  x  2 3  - x  1 3  x  2 4  .) n V(x 9  are  1  x  1  4>  x  2  3  a  n  d  X  24  + x )  n  1 4  x  V(x  )  2 3  3 4  n  V(x  3 4  )  ).  3 ) = JP where the homogeneous c o o r d i n a t e s 1»^  ( > ) n ( ) ^ 14 23— 13 24^ ^ 34^ 3 n V(x.„) i s a s u b v a r i e t y of JP . S i n c e x,,x.„ - x,„x„, does not i n v o l v e 12 14 23 13 24 x  3'  0, t h a t i s  J2(l,2) = V ( x  5 TP n V(x  us t h a t , on  a  n  d  s  o  n  x  2  v  x  =  1  V  X  X  X  X  n V  X  2  x_. or.x,„ we can w r i t e t h i s v a r i e t y V'(x.,x„„ - x , „ x „ . ) . T h i s i s simply the 34 12 14 23 13 24 1 1 ^ 3 image of JP x JP under the Segre embedding i n t o our c h o i c e . o f JP thus we have  (2.2.1)  £2(1,2) n V 0 x ) s l 2  On the other hand £2(1,2) n {X e JP ; 5  IP x JP 1  x  1 2  f 0} = £2(1,2)  neighbourhood o f (1,0,0,0,0,0) i n £2(1,2) and we the  a f f i n e c o o r d i n a t e s on W^ . 2  1  is a  can l o o k a t i t i n terms of  R e c a l l h e r e the d e f i n i t i o n . o f the s t a n d a r d  - 35 open cover of G^CE ) as defined i n Chapter I , sections 2 and 3, and s e t w For Xg W  = fi(.l,2)  W  n  .  the S t i e f e l matrix A, which represents X, such that A  = i 12 2  i s given by equation (.1.2.2) as fl  0  -x  -x l  2 3  2 4  A = ° where x^^ = ^ 3 2 4 ~ ]_4 23 X  X  X  X  13 ' 24' 14 ^ 2 3 4 X  X  anc  X  X  a S  a  ^  i  ^"  =  n  1  X  13  X  14  Since x ^ = det A^ = 1 we can consider 2  2  coordinates and as such  e  i s a quadric  hypersurface i n C . The dimension of the Z a r i s k i tangent space at the o r i g i n i s 4, but everywhere else i t i s 3 showing that (1,0,0,0,0,0)  i s an  i s o l a t e d s i n g u l a r point of £2(1,2). By a s i m i l a r analysis on each of the other open sets we can see that t h i s i s , i n f a c t , the only s i n g u l a r p o i n t , thus (2.2.2)  4 JP  Sing £2(1,2) = £2(0,0) .  For the remainder of':this discussion, then, we f i x the ambient space as 5 = IP n V ( x ^ ) . 4  We can now view £2(1,2) as the singular a f f i n e v a r i e t y  w^  completed by i t points at i n f i n i t y , i . e . those l y i n g i n the hyperplane x^  2  = 0.  As we have seen, the r e s t r i c t i o n of the Plucker embedding to t h i s  set of points at i n f i n i t y i s the same as the Segre embedding of JP x JP into 3 4 X  JP = JP n V ( x ^ ) . and since JP x JP n (2.2.3) X  X  Since £2(1,2) i s covered by the sets = 0 i t follows that JP x JP = U w.. V(x ). i=l,2 j=3,4 1  1  n  1 3  l 9  X  , open i n £2(1,2),  2>  -  36  -  T h i s l a s t u n i o n can he broken down more u s e f u l l y as  follows.:  (2.2.4)  ._V -w±j x x, z j=3,4  =  W  13  *  [ W  14  X  W  13  ]  U  [W  23  X W  13  J  1 1  [w  24  X  ( w  14  where \ denotes the s e t d i f f e r e n c e .  Relabel  D ,  To show t h a t  D  1  2  '  D  3  a  n  d  D  4  from l e f t  to r i g h t .  U W  23  ) J  the s e t s on the r i g h t ;as  4 i=l  is,  i n f a c t a d i s j o i n t u n i o n i t s u f f i c e s to show t h a t  D n x  but D  2  a point i n D  n D  3  n  1  must have x^  since x ^ x ^  =  Consider  x  x 1 3  A g a i n we the order  ij  be  must have £ 0 =f x^  D  3  =  0  = 0 = x^ and  x^  = 0.  and  x^  2  ^ 0, and  Both of these are  1  a point  in  impossible  4' then we  have  = (.C x C) XL ' ( W ' x C) XL (C x {«>}) i i { (oo,*,)} .  t h a t they appear.  (2.2.5)  A  = D2n  r e l a b e l the s e t s on the r i g h t hand s i d e as C, , -.C_, C_ and 1 2 3  decomposition  We  C, i n 4  wish to show that t h e r e e x i s t s such a  satisfying C.cD. x x  F i r s t we  4  P"^ = C XL'{:OO} , where oo i s the p o i n t a t i n f i n i t y  I P x TP 1  2  D  for  choose l o c a l c o o r d i n a t e s  i = 1,2,3,4. •? ? » on each w. . f o r i = 1,2 i j  and  j =. 3,4.  Let  xi the S t i e f e l m a t r i x of Xe w. . such t h a t the 2x 2 submatrix A., of  columns i and  j i s the i d e n t i t y , then, by analogy w i t h e q u a t i o n (1.2.2)  we  - 37 have,  :  A  1 3  =  '1 Q  Z  f  A  2 3  =  U  Z  0  l  V  1  l  1  3  u  X  U  V  2  0  Z  2  2 4  = Z  4  0 U  2  1  0  0  V  4  0' 1  0' 1  ' 13' 14' 23' 24 ' X  v =  1  X  X  )  i > r r iV'  ( z  x  u  v  u  v = ( z , u , 1, u v , v ) , 2  14 1 23 1 24  2  v = " 3>  U  v =  U  (  X  '1  3  X  12  x 13  X  =  l  A  1  where, s e t t i n g v = (  A  U  0 o •  0  3  A  z  ( _ z  2  2  2  3 ' 3 3' » 3 ' U  V  1  V  4 ' 4 4' 4' 4' V  U  V  }  1 }  Since the a f f i n e coordinates u,v and z on w „ are allowed to vary f r e e l y we see immediately that each w.. i s an a f f i n e space of dimension 3. I n p a r t i c u l a r we have dim  c  (fl(l,2)) = 3.  R e s t r i c t i n g the l o c a l coordinates to w!. = w.. .n V(x ) reveals how (2.2.5 ) ij i j 12 i s s a t i s f i e d . From (2.2.3) and (2.2.4) i t follows that IP  1  x W  1  = w|  3  n  IL [w]_ \ w j _ J 4  3  JJ, [ w  2 3  \w| J 3  j i {w\  D!, where D! = D. V ( x ) . . x x x 12 x=l n  n  \ (w] u  uw;,)J  "24 "14 " 2 3 ' fi  x v  A  - 38 Setting x  = 0 amounts to k i l l i n g  Ll  the z - c o o r d i n a t e on each w..,  In a d d i t i o n ,  ?i;  =  D! x  v  1  0  0  0  1  u  0  setting x  ; u v  e c > = c-x  = 0 amounts to k i l l i n g XJ  W  23'  s o  w  e  a  so we  have  I J '  x  s  o  n  a  v  2  the u - c o o r d i n a t e on w', and 14  e  1 D  c.  v  2  0  0  0  0  0  1  0  1 0  0  0  ; v  CX  e c  2  {co}  and 0 ;  Finally, setting  = x  c o o r d i n a t e s on w^.,  D  4  =  and  1  3  £  C  =  W x C ,  = 0 amounts to k i l l i n g both the u and  2 3  so the l a s t s e t i s  0  1  o o|  0  0  0  D! then i s the r e q u i r e d  v  v.  1  \ = {(»,»)}  s e t C. and we  have e s t a b l i s h e d  (2.2.5).  v  - 39 *-  Chapter I I I THE SCHUBERT CALCULUS §1  Intersection  Theory  I n t h i s s e c t i o n we summarize b r i e f l y t h e main i d e a s i n the t o p o l o g i c a l i n t e r s e c t i o n t h e o r y developed by L e f s c h e t z , A t the time o f w r i t i n g , t h e book P r i n c i p l e s o f A l g e b r a i c Geometry by G r i f f i t h s and H a r r i s has been recently published.  T h i s book c o n t a i n s a complete, u p - t o - d a t e v e r s i o n o f  t h i s t h e o r y , so any d e t a i l e d t r e a t m e n t h e r e would be redundant. oo  Throughout t h i s s e c t i o n M w i l l be a r e a l , o r i e n t e d C d i m e n s i o n m.  A singular p-chain C =  manifold of  on M, s a t i s f y i n g t h e p r o p e r t y  t h a t each s i n g u l a r p - s i m p l e x , r, , i s t h e r e s t r i c t i o n t o t h e s t a n d a r d p-simplex  A  p  c JR  of a C  CO  map from a neighbourhood o f A t o M i s c a l l e d P P a p i e c e w i s e smooth p - c h a i n on M. S i n c e t h e boundary o f a p i e c e w i s e smooth !  DS  c h a i n i s p i e c e w i s e smooth, we can d e f i n e a c h a i n complex i s a subcomplex o f t h e s i n g u l a r c h a i n complex.  C^  I t i s a fact  (M,Z) which from  d i f f e r e n t i a l t o p o l o g y t h a t t h e homologies o f t h e s e two c h a i n complexes are i s o m o r p h i c . An (m-p)-cycle and an (m-q)-cycle a r e s a i d t o i n t e r s e c t p r o p e r l y i f t h e i n t e r s e c t i o n has pure codimension p+q.  They a r e s a i d t o i n t e r s e c t  t r a n s v e r s e l y a t a p o i n t x i f t h e tangent space t o t h e i n t e r s e c t i o n a t x has codimension p+q a l s o . L e t A and B be two p i e c e w i s e smooth c y c l e s on M o f complementary A = p, dim B = m-p and l e t x e A n B be a p o i n t where JR JR A and B i n t e r s e c t t r a n s v e r s e l y , i , e v , t h e tangent spaces T ( A ) , T (B) t o A d i m e n s i o n s , i . e . dim  X  X  and B a t x a r e subspaces o f the tangent space T (M) and have dimensions p and m-p r e s p e c t i v e l y .  In fact  - .40 -  T CM) = T (A) f T (B) , X  X  X  L e t { u , , , , u } and {v.,,,,,v } be o r i e n t e d bases f o r T (A) and T (B) 1 p 1 m-p x x 1?  7  r e s p e c t i v e l y then we d e f i n e the i n t e r s e c t i o n index of A w i t h B a t x as follows det i  (u ,., . , ,u ,v ,,. , ,v ) .•• E_JL _JtP__ (u , , ,, ,u ,v ,,.,,v ) 1 p 1 m-p  (A,B) =  1  det  aat i s , + 1 a c c o r d i n g oriented basis f o r  t o whether o r n o t {u.,...,u ,v.,.,,,v } i s an Jp 1 m ^-p Jp 1 m«-D  ( M ) . In the case where A and B i n t e r s e c t t r a n s v e r s e l y  everywhere we d e f i n e  <A,B> =  I x (A,B) xeAnB x  and  call  justified  i t the i n t e r s e c t i o n number o f A w i t h B .  The word number i s  dim ( A n B) = 0 so A n B i s a d i s c r e t e subset  s i n c e , by h y p o t h e s i s ,  of M , which i s assumed t o be compact.  .Thus A n B i s f i n i t e .  One shows t h a t the i n t e r s e c t i o n number depends only on the homology c l a s s e s of A and B , t h a t i s i f A i s homologous t o A '  <A,B>  (3.1.1)  =  then  <A\B>,  or, s i n c e {,) i s b i l i n e a r ,  <A,B)  for  ^ o  any A homologous to zero. For  any two homology c l a s s e s a e H ( M , 2Z) and g s H ( M , 2Z) i t i s p o s s i b l e ^ P m-p  to f i n d two p i e c e w i s e smooth c y c l e s A and B r e p r e s e n t i n g  a and 3 r e s p e c t i v e l y  - 41 such that pairing  A a n d B meet t r a n s v e r s e l y  on A n B ,  So we c a n d e f i n e a b i l i n e a r  on homology  H (M, TL) x H (M, TL) + TL p m-p ' via  (a,S)  {a, B.)  +  which i s c a l l e d the i n t e r s e c t i o n theorem a s s e r t s  that  =  ( A , B ) ,  p a i r i n g o f a w i t h 3.  this pairing  i s unimodular, i n other words  Horn™ (H (M, TL) , TL) = { { a , ) iL m-p and  also  B sH  m-p  that  i f , f o r a f i x e d et.  H e  (M, TL) t h e n a i s a t o r s i o n  then  The P o i n c a r e  P  (  M  J  %•) ,  class.  duality  that  ; a e H (M, TL)} , P w  have ( a , g ) = 0 f o r a l l  e  I n p a r t i c u l a r i f H (M, TL) i s f r e e p r  i s non-degenerate. I n t h e c o u r s e o f p r o v i n g P o i n c a r e d u a l i t y we f i n d t h a t  there are  isomorphisms  D : H ( M , Q ) -y H p  m _ p  (M,4))  satisfying  (3.1.2)  (a.&)  =  [3,D(cO]  ^jhere [ , ] d e n o t e s t h e K r o n e c k e r p r o d u c t .  We w i l l  Q i s used as t h e c o e f f i c i e n t r i n g s i m p l y t o k i l l the  next s e c t i o n  the torsion.  later.  I n view of  t o present the main p o i n t s of t h i s s e c t i o n .  t h i s e n d l e t M now b e a c o m p a c t , c o m p l e x m a n i f o l d  p = 2 3.  (3.1.2) u s e f u l  this i s purely a technicality,  We a r e now i n a p o s i t i o n  m = 2n.  find  L e t V be an a n a l y t i c L e t V have i t s n a t u r a l  To  o f d i m e n s i o n n o v e r <E, s o  s u b v a r i e t y o f M o f c o m p l e x d i m e n s i o n 6, a n d s e t orientation.  We c a n a s s i g n a c o h o m o l o g y  class  - 42 to V i n the f o l l o w i n g way.  Let a e  p^s> ^  a n d  choose a representative  cycle A that meets V transversely i n smooth p o i n t s ,  Again one shows that  the i n t e r s e c t i o n number <V,A> =  I i (V,A) xeAnV  i s independent of the choice of representative A f o r a, but t h i s i s not as straightforward as i n the case of (3.1.1) due to the p o s s i b i l i t y of s i n g u l a r i t i e s on V, suppose A and A  1  We get around t h i s problem by using the f o l l o w i n g t r i c k ;  are both representatives of a, then, using the f a c t that the  singular locus of V i s a proper subvariety of V, hence has r e a l codimension > 2, i t i s p o s s i b l e to f i n d an m-p+1  chain C on M which does not i n t e r s e c t  the s i n g u l a r locus of V, which meets V transversely almost everywhere and has the property that 3C = A-A' . For the remainder of the proof one proceeds as one would do i n the case of (3.1.1).  For t h i s , the interested reader i s r e f e r r e d to G r i f f i t h s ' and  H a r r i s ' book.  V then defines a l i n e a r f u n c t i o n a l H m-p  Poincare' d u a l i t y i s of the form i t s Poincare dual D(y) g H  m  (M, ZZ) -+TL, which by ' J  } f o r some yeH^(M, ZZ.) . This c l a s s or  ^(M, ZZ) i s c a l l e d the fundamental class of V.  One might ask i f a l l t h i s i s necessary, f o r a simpler way of assigning a cohomology c l a s s to V would be to look f o r a submanifold V' homologous to V, then assign V the fundamental class of V'. not p o s s i b l e .  However t h i s , i n general, i s  I t i s proved i n the paper Nonsmoothing of a l g e b r a i c cycles  on Grassmannian v a r i e t i e s by Hartshorne, Rees and Thomas that the Schubert v a r i e t y £2 (.2,2,3) i n G (E^.) i s not " t o p o l o g i c a l l y smoothable," Q  - 43  -  Suppose W i s an a n a l y t i c s u b v a r i e t y of M h a v i n g complex dimension n and  intersecting V transversely  the  orientations  of V , W and  l  This f a c t i s c e n t r a l between r e a l and  x  (.V,W) =  to a l g e b r a i c  c r o s s another c y c l e , cancellation.  1,  It r e f l e c t s a basic  I f V and  i n t e r s e c t i o n number i s the  number of  change d i r e c t i o n and  c a r d i n a l i t y of  the  suppose that V and  count the variety  W do  e  C  n  3  not  of dimension m w h i c h i i s  c h o i c e of l o c a l c o o r d i n a t e s ,  i t i s possible  xeVnW  have  can  analytic  Here we  t h i n k of  u  an  have  W at x,. and  in m  (V,W)  does not  W intersect in a  e x p l i c i t l o c a l analysis  i Thus we  I f V and  to f i n d a r e p r e s e n t a t i v e W'  which meets V t r a n s v e r s e l y  I  f i n d i n g an  i n t e r s e c t i o n V n W + e.  i n t e r s e c t i o n m u l t i p l i c i t y of V and  number of p o i n t s then, by  x.,  W at a p o i n t  v,W.  i s c a l l e d the  x^,  Essentially  a u - s h e e t e d branched cover of a neighbourhood  x  the  usually  i t s homology c l a s s we  T h i s i s done l o c a l l y by  m (V,W) =  on  transversely.  to V and  perturbing W s l i g h t l y inside  l o s t tangents.  may  intersection.  intersect  some tangent d i r e c t i o n s  of x whose f i b e r over e i s the £  points  c r o s s back p r o d u c i n g a  somewhat s m a l l e r than the  By  in a  i n t e r s e c t i o n , whereas i n the r e a l case a c y c l e  i n t e r s e c t i o n number i s  coalesced.  difference  W intersect transversely  r e a l case, the  t h i s means that  choice f o r  that  i n the  Now  So,  easily  geometry.  complex geometry,  set-theoretic  S i n c e there i s a n a t u r a l  M i t follows  f i n i t e s e t of p o i n t s then the i n the  at x,  ^9  at each of  depend  finite the  points  of the homology c l a s s of  p o i n t s i n the  W  chosen neighbourhood  of  (v,w) = {v,v:»}  I  m  x. eVfiw  (V,W) ,  ±  X  We note t h a t a t r a n s v e r s e i n t e r s e c t i o n p o i n t i s c h a r a c t e r i z e d by h a v i n g unit  multiplicity, I t i s a s i m p l e m a t t e r now t o see t h a t t h e i n t e r s e c t i o n p a i r i n g i s  P o i n c a r e d u a l t o cup p r o d u c t i n cohomology, H  m  P  Consider the f o l l o w i n g  diagram  (M,U)) x H (M,q) P  ([M]n-) x i d  H (M,<Q) x H (M,«}) P  K  p  J  id. x D  H (M,OJ) x H (M.Q)p m-p where D i s t h e P o i n c a r e d u a l i t y isomorphism, I the i n t e r s e c t i o n p a i r i n g and K the K r o n e c k e r p r o d u c t .  C o m m u t a t i v i t y o f t h e lower t r i a n g l e f o l l o w s from  (3.1.2) and t h e f a c t t h a t f o r a n a l y t i c c y c l e s v e H (M, ZZ) , w e H P J  1- (V,W) X  (M, ZZ) m -  P  = l (w,v) = +1 x  consequently (v,w) = (w,v). [M] i s t h e fundamental c l a s s o f M and n denotes the cap p r o d u c t so [M] n an a l t e r n a t i v e d e s c r i p t i o n o f t h e P o i n c a r e d u a l i t y isomorphism.  i  S i n c e cup  and cap p r o d u c t s a r e a d j o i n t w i t h r e s p e c t t o t h e K r o n e c k e r p r o d u c t , c o m m u t a t i v i t y o f the upper t r i a n g l e f o l l o w s where C denotes t h e cup p r o d u c t .  - 45 I n case ex,(3 a r e two homology c l a s s e s not of complementary d i m e n s i p n we  can s t i l l d e f i n e an i n t e r s e c t i o n p r o d u c t .  Suppose a e H (M, 2Z) • m-p (M, ZZ) , t h e r e e x i s t r e p r e s e n t a t i v e s A and B i n t e r s e c t i n g  g e  t r a n s v e r s e l y almost everywhere, that i f { v . . , . . . , V 1  We  and  choose the o r i e n t a t i o n f o r C = A n B  so  } i s an o r i e n t e d b a s i s f o r T (C) on smooth p o i n t s x  m-p-  q  x,  and i f  {  { v  l""'V l""'Vp-q  U  V  l""' m-p- ' l'"" p V  W  W  }  and  }  q  are bases f o r T (A) and  {  U  T (B) r e s p e c t i v e l y then  1" ' '' q'V  • • " m-p-q' l' ' ' * ' p  U  V  i s an o r i e n t e d b a s i s f o r T (M). and we denote i t A - B ,  One  W  W  }  C i s then c a l l e d the i n t e r s e c t i o n c y c l e  checks t h a t t h i s p r o d u c t i s w e l l - d e f i n e d  on  homology by f i n d i n g a c h a i n D i n t e r s e c t i n g B t r a n s v e r s e l y almost everywhere such t h a t 3D = A and  then showing t h a t  3(D * B) = A • B  h o l d s when O r i e n t a t i o n s a r e t a k e n i n t o a c c o u n t . product i s a s s o c i a t i v e ,  T h i s p r o d u c t and  One  a l s o checks t h a t  the  i n d e e d a l l c o f the p r e c e e d i n g i s  c e n t r a l to the j u s t i f i c a t i o n of the Schubert c a l c u l u s .  - 46 §2  The Grassmannian  as C.W, Complex  The a p p l i c a t i o n of an i n t e r s e c t i o n t h e o r y t o the s u b v a r i e t i e s o f the Grassmannian T h i s was  would be g r e a t l y ^ e n h a n c e d by an e x p l i c i t b a s i s f o r the homology.  f i r s t d e s c r i b e d by Ehresmann who showed t h a t the Schubert v a r i e t i e s  ( o r more p r e c i s e l y t h e i r i n t e r i o r s ) p r o v i d e a c e l l d e c o m p o s i t i o n o f ^ ( E ^ ) . R e c a l l the d i s c u s s i o n i n the f i r s t s e c t i o n of c h a p t e r one.  There we saw  t h a t every k-plane X has a unique i n t e r s e c t i o n sequence i ( X ) which i n t u r n corresponds to a unique Schubert symbol, t h a t i s the sequence (a^, . . . ,a^_) of l a g s i n the i n t e r s e c t i o n sequence.  T h e r e f o r e G. (E ) i s decomposed as a rC  d i s j o i n t u n i o n of t h e s e t s u(a^,...,a ) k  = {XeG^XE^)  (a^,...,a^)  ; ( a ^ , . . . , a ) i s the l a g sequence o f i ( X ) } n  = {XeG. (E ) ; dim (XnE ) = i, K n a.+l l T h i s s e t i s the complement i n Schubert s u b v a r i e t i e s .  XI  where  ft(a^,....a^)  dim (XnE . . ) = i-1} . a.+i-l '' l T  o f the u n i o n of a l l i t s p r o p e r  By c h o o s i n g these s u b v a r i e t i e s as l a r g e as p o s s i b l e  i n s i d e n(a^,...,a^)- we can show the f o l l o w i n g ; Proposition  (3.2.1)  a ) = JUa^ . . . ,a ) \ [ I fc  fc  fi(a  a^a^.-l,...,^)  1  ieN  *  where N = {1 < i < k; a. > a. ,} and where we s e t a = 0. I i - i o Proof I f b. < a. and dim (XnE, ,.) > i then dim (XnE ,.) > i , thus i f t h i s i i D .+i a ,+i I  I  h o l d s f o r 1 < i < k thenft(b., . ,, ,b. ) cft( ., , ,, ,a, ) . 1 k 1' k decomposable k - y e c t o r  But i f b. < a. then the i i  a  —  e  1  n  A  e  2  A . . . A  e . , , \ e, , . , A i-(a.-b.) b.+i+l i i i A  N  . , . A  e  ,. A , a.+i I  , , A  e  a.+k l  - 47 s a t i s f i e s dim ( X n E , . ) =? i a. T I  dim ( X n E, , .) > i , b.t i  but not  l  fi(b  1?  , , , b ) c. ?  fi(a ,  k  1?  I t follows that  l  , , ,a )< = >b k  i  < a,  1 < i < k  ±  and t h a t the containment i s proper e x a c t l y when a t l e a s t one o f these inequalities i s strict.  Thus u ( a ,...,a,) i s non v o i d , X  K.  Suppose X e oo(a , , ,, ,a ) then dim ( X n E X  K.  n  cl  ,Tl  X £ tt(a^,, . , . a ^ a - 1 , . . .a^) = £ 2 X f o r a l l i ,  ) = i-1 < i  thus  ^X  Conversely i f X k £ 2 1 , f o r  A  a l l i , then dim (X n E  < 1.  &  has jumps o f a t most one.  However i t i s a p r o p e r t y o f i ( X ) t h a t i t  Thus dim (X n E  , . - , ) = i - 1 and X e £2(a.. , . . . ,a, ) i  q.e.d. Let  us c o n s i d e r f o r a moment t h e example o f £2(1,2) c G^CE^).  (3.1.2)  t e l l s us t h a t a ) ( i , 2 ) = £2(1,2) \ (£2(1,1) un(o,2)). £2(1,2) i s d e f i n e d as a s u b v a r i e t y o f G2(E^) by s e t t i n g t h e P l u c k e r c o o r d i n a t e £2(1,1) and £2(0,2) have t h e a d d i t i o n a l r e l a t i o n s x ^ = x^^ = 0 and  x^  = 0.  x  = x„. = 0 r e s p e c t i v e l y by ( 2 . 1 . 2 ) . 24  23 0 0  J  S i n c e x_,x„„ = x-_x„, i t i s n o t 14 23 13 24  p o s s i b l e t o v i o l a t e the c o n d i t i o n s o f membership i n £2(1,1) and £2(0,2) s i m u l t a n e o u s l y by h a v i n g x ^ ^ 0 4o n l y i f x^^ 4 0.  and x^^ = 0.  We have then 03(1,2) = £2(1,2) n W  which g e n e r a l i z e s to  2  4  Thus X eo)(l,2) i f and  - 48 -  Proposition  (3,2,2)  U  (.a  , , . ,a ) = B C a  1 ?  k  . ., ^  l f  nW  +  1  '  k  '* k  Proof A p o i n t i n £2^ s a t i s f i e s c e r t a i n l i n e a r r e l a t i o n s a c c o r d i n g t o ( 2 . 1 . 2 ) . Some o f t h e s e a r e the d e f i n i n g r e l a t i o n s f o r £2(a^,, . , ,a^) , b u t t h e s e t o f e x t r a El'ucker c o o r d i n a t e s t h a t v a n i s h on S2 i s e a s i l y seen t o be 1  R. = { x . . ; j . = a.+i and i J-L* -. . »J i i k  Since x  f o r X f i} .  j , < a, + A > X  e| R. i t f o l l o w s t h a t a k-plane X i n W ,.. a.,+1, . . . ,a, +k . x a.,+1, ... ,a,+k 1 k ien 1 k  cannot be i n any £2 , thus 1  0  (  a  l V  0  W  a +l ... a +k f  1  f  £uCa  k  r--- k a  )  I f X e u> (a, , . ..,a, ) then dim ( X n E , .) = i and dim ( X n E , . . , ) = 1 k a.+i a.+x-l I  i - 1,  I  so we can choose a b a s i s f o r X such t h a t the i ' *  1  v e c t o r has a.+i "' 1  1  l  c o o r d i n a t e non-zero.  T h e r e f o r e a S t i e f e l m a t r i x A, f o r X d i f f e r s from t h a t  i n t h e p r o o f o f (2.1.2) o n l y i n t h a t each x . X *  The k * k s u b m a t r i x A  .  3 ^ T I j • • • , a ^ t K  . i s guaranteed non-zero. Si 1  . 'X  i s a lower t r i a n g u l a r m a t r i x w i t h no z e r o  e n t r y on t h e d i a g o n a l , thus P  a  n  d  X e W  a  1 +  a  1 +  l,...,a +k  =  ( X )  k  l,...,a +k' k  d  e  t  \  +  l  * °  ^ ' " e  d  We have a l r e a d y seen i n c h a p t e r 2, s e c t i o n 2 t h a t £2(1,2) nW.. a f f i n e space o f d i m e n s i o n 3 over t h e complex .numbers, a homeomorphism  i s an  In p a r t i c u l a r there i s  - 49 -  (0(1,2) £  e x h i b i t i n g to(1,2) as a 6 - c e l l ,  JR  6  This also generalizes to  P r o p o s i t i o n (3.2.3) 2(a +,.,+a ) k  u ( a ^ , . , . , a ) i s homeomorphic t o 3R k  Proof The S t i e f e l m a t r i x A o f X mentioned i n t h e p r o o f o f (3.2.2) has a. priori  k £  a.+i e n t r i e s t h a t a r e u n s p e c i f i e d b u t A i s n o t a unique  r e p r e s e n t a t i v e o f X.  I f we choose A i n s t e a d so t h a t A  ,, = a^+1,..., k+k a  I k  as u s u a l , o r e q u i v a l e n t l y choose  ^ then we s p e c i f y an e x t r a  ^a^+1,...,a^+k ^ -^(k+l) e n t r i e s c o r r e s p o n d i n g t o those on and below  the d i a g o n a l i n A . There remain e x a c t l y a,+..,+a, e n t r i e s i n a^+1,..., +k 1 k M  a  k  A' t h a t a r e f r e e t o v a r y over  C.  q.e.d.  From t h i s i t f o l l o w s e a s i l y t h a t ti(a^, . . . ,a^_) i s i r r e d u c i b l e s i n c e i t i s t h e c l o s u r e o f o i ( a ^ , . . . ,a^) w h i c h i s c o n n e c t e d . So t h e Grassmannian G, (E ) i s t h e d i s j o i n t u n i o n o f a f i n i t e number o f k n even d i m e n s i o n a l c e l l s , and by (3.2.1) t h e c l o s u r e o f any c e l l w (a. ,.. . ,a,) J  i s contained  i n the 2(a,+. . .+a, )^-skeleton. 1 k  Thus G, (E ) i s a f i n i t e CW k n  complex, CW The CW c h a i n group  ( G ^ ( E ^ ) , ZZ) i s t h e f r e e a b e l i a n group on a l l t h e  c e l l s w(a^,.,,,a^) such t h a t a^+ , + dimensional  = r . S i n c e a l l t h e c e l l s a r e even  every c h a i n i s a c y c l e and no c h a i n i s a boundary, so t h e  homology groups a r e n a t u r a l l y  i s o m o r p h i c t o t h e c h a i n groups,  A homology c l a s s depends o n l y on t h e sequence o f i n t e g e r s s i n c e any two f u l l f l a g s  E , E ' o f l i n e a r subspaces o f  (a^  t l t f  a^)  ( i . e . any two  bases f o r E ) a r e connected by an i n v e r t i b l e l i n e a r t r a n s f o r m a t i o n from n •> ~ E^ t o i t s e l f which i n d u c e s an i n v e r t i b l e , p r o j e c t i v e l i n e a r t r a n s f o r m a t i o n from  J P ( A E ) t o i t s e l f c a r r y i n g o)(.a , , ,, ,a ) , d e f i n e d r e l a t i v e t o E, n x ic  b i j e c t i v e l y onto ' ( a ^ j i ' t > ^ ) »  defined r e l a t i v e to E',  u  We have p r o v e d  The B a s i s Theorem, P a r t I The i n t e g r a l homology o f G (E ) i s f r e e l y g e n e r a t e d i n d i m e n s i o n 2r by JC n the Schubert symbols (a ,.,.,a ) where a +...+a = r . The o d d - d i m e n s i o n a l X  K.  X  K.  groups a r e a l l z e r o . So, f o r example, t h e homology groups o f G^CE^) t h a t a r e n o n - t r i v i a l have one g e n e r a t o r ( 0 , 0 ) , ( 0 , 1 ) , (1,2) and (2,2) i n dimensions 0,2,6 and 8 r e s p e c t i v e l y and two g e n e r a t o r s (0,2) and (1,1) i n d i m e n s i o n 4. F i n d i n g t h e B e l t i numbers o f G, (E ) i s a c o m b i n a t o r i a l problem. k. n  The  th 2r B e l t i number, g i s t h e number o f S c h u b e r t c y c l e s ( a ,...,a ) such t h a t a^+a +...+a^ = r . I f r < k then t h i s i s t h e number b ( r ) o f p a r t i t i o n s 2  of r . When r > k t h e s i t u a t i o n i s n o t q u i t e so s i m p l e , b u t we c a n s i m p l i f y as f o l l o w s , l e t N = d i m ( j ( G  c  ))  E n  k ( n - k ) , we show t h a t g  =  = 2 r  $2(N-r) "  L e t S, be t h e s e t o f Schubert symbols d e f i n e d over G, (E ) then t h e f u n c t i o n k,n k n f :S  -y S  iC f TL  d e f i n e d by s e n d i n g t h e symbol ( a , . . . , a ) t o t h e symbol 1  IC y n  X  (n-k-a^,...,n-k-a^)  iC  i s a b i j e c t i o n w h i c h r e s t r i c t s t o a b i j e c t i o n between  the s e t o f symbols w i t h sum r and t h e s e t o f symbols w i t h sum k(n^k) - ( a + ,..+a,) = N - r . "1 k n  let N  T  Thus we have shown t h a t B „ = B „ / X T . 2r 2(N-r) x  = the greatest Integer i n  then f o r k < r < N '  of p a r t i t i o n s of r i n t o a t most k p l a c e s , N-r <_ N' and B 2 r  =  ^2(N-r)  S^  Yes  u  s  those.  B2r  =  t  n  e  Now number  F i n a l l y , i f N' < r < N then  - 51 S i n c e G, (E ) i s a compact, complex m a n i f o l d t h e statement k n • ' f o l l o w s more g e n e r a l l y from P o i n c a r e d u a l i t y . t h a t i f [ J i s theccohomology  3  n  2r  = 3„  s /lT 2(N-r)  I n f a c t we s h a l l see l a t e r  c l a s s of t h e Schubert symbol  and [ ]  denotes the d u a l c l a s s then  [ ]* - If(  §3.  )]  The R i n g S t r u c t u r e i n Homology I n t h e case where t h e f i g u r e s a r e l i n e a r spaces an enumerative  problem  can, t h e o r e t i c a l l y , always be s o l v e d u s i n g t h e r e s u l t s o f t h e p r e v i o u s two sections.  F i r s t t h e s o l u t i o n s e t i s d e s c r i b e d as a z e r o - d i m e n s i o n a l sub-  v a r i e t y o f the Grassmannian,  t h a t i s , w r i t t e n as an i n t e r s e c t i o n o f a f i n i t e  number o f Schubert v a r i e t i e s ( i n t e r s e c t i n g i n a f i n i t e number of p o i n t s ) . S i n c e t h e d e s c r i p t i o n o f t h e s o l u t i o n s e t u s u a l l y i n v o l v e s some d e g e n e r a t i o n i n t o s p e c i a l p o s i t i o n these p o i n t s a r e counted w i t h m u l t i p l i c i t i e s .  We have  seen t h a t t h e i n t e r s e c t i o n number t h e n , counts t h e number o f s o l u t i o n s i n the g e n e r a l case s i n c e " s p e c i a l i z i n g " can be i n t e r p r e t e d as moving w i t h i n a homology c l a s s . The problem i s t o f i n d t h e p r o d u c t o f two Schubert c y c l e s as a l i n e a r c o m b i n a t i o n of o t h e r Schubert c y c l e s .  The f i r s t s t e p i s t o f i n d t h e  i n t e r s e c t i o n numbers o f p a i r s o f c y c l e s i n complementary  dimensions.  To t h i s  end l e t £2 (a.. , . .., a, ) and £2 (b, , . . . ,b, ) be Schubert v a r i e t i e s i n G, (E ) so t h a t l k l k k n k k I b. = k ( n - k ) - I 1=1 1=1 1  a. . 1  These v a r i e t i e s do n o t , i n g e n e r a l , i n t e r s e c t t r a n s v e r s e l y ,  For consider  £2(1,2) and £2(0,1) i n G ( E ) , We have codim £2(1,2) = 1, codim £2(0,1) = 3, 2  4  I t f o l l o w s d i r e c t l y from t h e Schubert c o n d i t i o n s t h a t  - 52 £2(1,2) n £2(0,1) = £2(0,1) and so codim (£2 (1, 2) nft(0,1)) < codim (£2(1,2)) + codlm ( £ 2 ( 0 , 1 ) ) , be n e c e s s a r y then t o f i n d a d i f f e r e n t r e p r e s e n t a t i v e  ft'(b^,,,,,b^)  It will f o r the  c l a s s o fft(b^, .,, j b ^ ) s o - t h a t t h e i n t e r s e c t i o n £2(.a^, , , , ,-a^) nft' (b^, , . , ,b^) is transverse.  We r e c a l l t h e f l a g  E = (E. 0  c  E  c ...  1  c  E ) n  d e f i n e d i n Chapter 2, s e c t i o n 1, where we chose {e^,...,e^} as a b a s i s f o r E^, and we d e f i n e a f l a g  =  3E'  ( E ' c E ' c  0  . . . c E ' )  1  n  where E'. i s t h e l i n e a r span o f {e x n-x+1 s t i l l denotes t h e d i m e n s i o n . fl' (b. , .. . ,b. 1 k and c o n s i d e r it  Now  ,e }, so t h a t t h e s u b s c r i p t n  define  = {X G, (E ) ; dim (X n E' .) > i , k n b.+x — x  )  e  fi(a^,...,a^)  nft'(b^,. .. ,b^) .  1 < i < k} — —  I f X i s i n t h i s i n t e r s e c t i o n then  satisfies  dxm ( X n E dim  , .) > x a.+x — x  (XnE'  and  .) > i ,  D .t l X  for 1 < i < k .  —  —  —  A more c o n v e n i e n t way o f w r i t i n g t h e second c o n d i t i o n i s dim  ) , k - i + 1,  (XnEk-x+1  Combining t h e two c o n d i t i o n s i t f o l l o w s t h a t  (3.3.1)  XnE  _.  „E» x  k-x+1  + 1 ?  M0),  i s i s k  53  T  -  since Y=  IXnE  ] + [XHE-  a  ]  i thus: Y has  d i m e n s i o n a t most k«  E and  force  E  T  S  X  k-i+1  a.+i + b. 1  C3.3.1) to h o l d , the d e f i n i t i o n s of  For  .,.,+k-i+1 > n + 1 k-i+1  that i s  a. +b. . > l k-i+1  arid by v i r t u e of the f a c t t h a t  n-k, '  ft(a_,, .,a,) f  and ft (h ,. . . ,K ) a r e  1'  k  1  of  k  complementary d intension : t h i s becomes a.+b, l We  .,. k-i+1  =n-k.  have thus shown t h a t s H a ^ , . . . ^ ) n fi'-(b , .. . ,b ) fc  b i. = n - k - a.k-i+1 I  a n d  i n  t  ^  l i s  c  a  s  e  i s empty u n l e s s  f i n d the i n t e r s e c t i o n e x p l i c i t l y .  w e  We  have  E  a.+i " b _. +k-i+l E  k  which i s simply  j . i »  a,+1  .1  e  a„+z  + 1  =  E  a.+i " n+l-(a.+l)'  the l i n e a r span o f e  e  ,i  E  ... a .+i l  By  (3.3.1) i t f o l l o w s t h a t  ) i s a b a s i s f o r X and c o n s e q u e n t l y dim ( X n E  a,+k  ^  J  ,.) a.+i  is exactly i .  T h i s p u t s X i n coCa.^, , , . ,a )  ft(a^, , , , ,a ) ,  L e t t i n g "{v^, , , ,v^_} denote the l i n e a r span o f y^, , , , ,v_.,  k  find  that  k  t  so X i s a smooth p o i n t  of  - 54 dim ( X n E ' ,. = dim ( X p E ' ,  V  n  1  )  ' 'Vi i k  +1  +  =  1  and so X i s a l s o a smooth p o i n t o f 0,' (b , . , , ,b, ) . X  intersection i s transverse.  By d e n o t i n g  One shows t h a t the  K.  the c l a s s o f n ( a ,. ,.,a,) by the x  tc  Schubert symbol (a , ...,a,) we have shown t h a t X  K.  {(a ,...,a ),(b ...,b )} ±  k  15  where I = ( n - k - a , n - k - a _ ^ , k  the c o n t e n t  =  k  6*  n-k-a^) and J = ( b ^ , . . . , b ) .  k  This i s  k  of  The B a s i s Theorem, P a r t I I  (a^,...,a )  = (n-k-a  k  k >  n-k-a^^,  n-k-a^)  * where  denotes t h e P o i n c a r e  dual.  T h i s theorem a l l o w s us t o f i n d t h e " c o o r d i n a t e s " o f an a r b i t r a r y (or i t s homology  c l a s s ) r e l a t i v e t o the b a s i s f o r t h e 2 r ^ homology group. t  L e t a be any c l a s s i n H  a  "  (G (E ) , ZZ), then by p a r t one o f the b a s i s theorem  I a ,,..,a 6  1  ( a k  l'"" k a  )  where t h e sum runs over a l l Schubert symbols such t h a t a^+,, +a = r . P  6'-  2r-cycle  k  To f i n d  we i n t e r s e c t both s i d e s w i t h (n-k-a , , , , ,n-k^a ) whose i n t e r s e c t i o n 1  - 55 number i s one w i t h ( a ^ , , , , a ^ ) and z e r o w i t h any o t h e r c l a s s i n the ?  sum.  That i s  7 a , ( n - k - a ,,,«,n-k-a ) \ = §  The i n t e g e r s S  a  1  , , . . , a  the case t h a t a was Grassmannian.  were r e f e r r e d to by Schubert as the degrees o f V i n  k  the c l a s s of an i r r e d u c i b l e s u b v a r i e t y V of the  Suppose W i s another i r r e d u c i b l e s u b v a r i e t y of  d i m e n s i o n and l e t y,  ,  "•^ > • •  be i t s degrees, t h a t i s  •»  •^  [ W J  complementary  ---» k b  b i >  ( b l  '""  , b k )  where [W] denotes the homology c l a s s o f W and the sum ranges over such t h a t b,+. . .+b, = N - r . 1 k  cL  • • •  Then we have  cl  1''*"' k  n-k-a^,...,n-k-a^  where a g a i n the sum ranges over a l l Schubert symbols r.  I n the case where k = l i . e . G, (E ) = k n  Bezout's  sequences  P  n  (a^,...,a ) i n dimension k  ^ t h i s e q u a t i o n reduces to  theorem.  The f o u n d a t i o n s o f the Schubert c a l c u l u s a r e s e t down w i t h the b a s i s theorem t o g e t h e r w i t h the f o l l o w i n g two theorems which a l l o w us to compute the p r o d u c t of two a r b i t r a r y Schubert c y c l e s i n terms o f a p a r t i c u l a r s e t c a l l e d the s p e c i a l Schubert c y c l e s .  Let  56  -  -  a'(d) = n ( n - k - 1 , , , , , n - k - l , n - k , , , , , n - k ) where t h e number o f ( n - k ^ l ) ' e t h a t appear i s e x a c t l y  d, and l e t c ^ denote  til  the homology c l a s s o f a ( d ) ,  a ( d ) i s c a l l e d the d  s p e c i a l Schubert  v a r i e t y and we can now s t a t e P i e r i ' s Formula  ( a ^ .. .,a ) • o k  d  = £(1^, .. . ,b ) fc  where t h e sum ranges over a l l Schubert symbols  ( b , ... ,b ) s a t i s f y i n g 1  a. .. < b. < a. f o r 1 < i < k and s a t i s f y i n g l-l — 1 — 1 — — J to codim - (b.. ( L l  b, ) = codim„ (a..,...,a.) + codim., a K.  (  L  l  <L a  ; lc  k ' where c o d i n g (\^,...,\^) = k(n-k) -  J X.. i=l  To i l l u s t r a t e t h i s we compute the s e l f i n t e r s e c t i o n o f (1,2) eH (G (E ), 7L). 6  coding  2  4  = 1.  By d e f i n i t i o n (1,2) = a , and codim 2  (1,2) =  (1,1) and (0,2) a r e the o n l y Scbubert symbols w i t h t h e d e s i r e d  codimension and they b o t h s a t i s f y the i n e q u a l i t i e s r e q u i r e d  by P i e r i ' s f o r m u l a .  Hence  (3.3.2)  (1,2) (1,2) = (1,2)  a  i  = (1,1) +  (0,2).  There i s a companion t o P i e r i ' s f o r m u l a which shows t h a t H ( G ( E ) , 2Z) A  can be g e n e r a t e d as a r i n g by t h e c l a s s e s i n t e r s e c t i o n c y c l e as the p r o d u c t ,  k  n  o f s p e c i a l Schubert c y c l e s , w i t h t h e  It i s  - 57 ~ a  a  a  1 a  Ca.1' "'*' k^ a  a  2  a,+k-2 k  a. l  where  a, k  i s d e f i n e d t o be z e r o i f i < 0 o r i > k.  Both o f t h e s e theorems and H a r r i s .  a r e proved complex a n a l y t i c a l l y i n G r i f f i t h s  Their: t r e a t m e n t would be d i f f i c u l t t o improve upon so we  r e f r a i n from r e p r o d u c i n g t h e p r o o f s h e r e . L e t us a p p l y o u r new found t e c h n i q u e s t o t h e s i m p l e e n u m e r a t i v e problem of f i n d i n g t h e number o f l i n e s s i m u l t a n e o u s l y meeting f o u r g i v e n l i n e s i n 3 g e n e r a l p o s i t i o n i n TP . We have a l r e a d y seen t h a t t h e s o l u t i o n s a r e t h e points i n the subvariety 4  v = n i=i  ^[L,,JP ] 3  3 of G^(TP ) = G^iE^), where L^,L ,L^ 2  and L^ a r e t h e f o u r l i n e s .  Now  3 ii[L^,TP  ] i s t h e same v a r i e t y as £2(1,2), by t h e remark f o l l o w i n g  where L_^ = TP(E ^) and TP  5  = TP (E^) .  Consequently i ,  (2.1.2),  t h e number o f p o i n t s i n  V counted w i t h t h e i r m u l t i p l i c i t i e s , i s t h e i n t e r s e c t i o n number o f t h e f o u r f o l d s e l f - i n t e r s e c t i o n o f (1,2), t h a t i s i = ( 1 , 2 ) « 4 C 1 , 2 ) , C 1 , 2 ) ) ' = -<(1,1) + CP,2),C1,1) + C 0 , 2 ) } , 4  2  2  The l a s t e x p r e s s i o n f o l l o w s from (3,3.2) and can be computed as f o l l o w s ;  58  -  -  {(1,1) + (0,2),(1,1) + (0,2)}  =  <(1,1),(1,1)>  + l{ ( 1 , 1 ) , (0,2),} + { ( 0 , 2 ) , (0,2)}  and t h e second p a r t o f t h e b a s i s theorem a p p l i e s k i l l i n g  t h e second  term,  w h i l e the f i r s t and l a s t terms a r e b o t h 1 s i n c e (1,1) and (0,2) a r e P o i n c a r e self-dual.  I n agreement w i t h our p r e v i o u s s o l u t i o n s o f t h i s problem we  find  t h a t t h e r e a r e two l i n e s i n g e n e r a l i n t e r s e c t i n g f o u r g i v e n l i n e s i n g e n e r a l p o s i t i o n i n complex p r o j e c t i v e Let of  3-space.  us compute a h i g h e r d i m e n s i o n a l example.  We w i s h t o f i n d the number  2-planes i n p r o j e c t i v e 5-space t h a t i n t e r s e c t 9 g i v e n 2 - p l a n e s .  c o n d i t i o n t h a t a 2-plane X meet a g i v e n 2-plane f[  dim (X H ) n  but s i n c e dim (X n I P ) = 2 , 5  The  is  > 0  2  then dim (X n TI-j)  S 0 and dim (X n TJ.^) > 1 by a  p r o p e r t y of i n t e r s e c t i o n sequences, where ™ 5 n  and dim n_  = i .  2 - 3 - 4 n  Thus X l i e s  '  n  i n the Schubert v a r i e t y fl[n ,n , 2  TP ] 5  4  = fiflP (E ),TP (E ),TP ( E ) ] 3  5  6  = £2(2,3,3) which i s t h e f i r s t s p e c i a l Schubert variety on G^CE^) = G (JP"'), 2  The number  we w i s h t o compute i s e q u a l t o the n i n e - f o l d s e l f i n t e r s e c t i o n number of the s p e c i a l Schubert c y c l e  = (2,3,3) ,  A p p l y i n g P i e r i ' s formula,, the square of  is (1,3,3) + (2,2,3) = o + c 2  where a  2  denotes t h e c y c l e ( 1 , 3 , 3 ) ,  h  CT  '  i  +  a  2  The t h i r d power of  is  2 • °i  which by P i e r i ' s f o r m u l a i s  (0,3,3) + 2(1,2,3) + ( 2 , 2 , 2 ) . R e c u r s i v e l y we  find  o j = 3(0,2,3) + 2(1,1,3) + 3(1,2,2) and  a\ = 5(0,1,3) + 6(0,2,2) + 5 ( 1 , 1 , 2 ) . Now by p a r t I I of the b a s i s theorem we can e v a l u a t e  °1  =  °1  ' °1  =  3  X  5  +  2  x  6  +  3 x 5 = 42.  As mentioned i n the i n t r o d u c t i o n , i t has been shown, i n g e n e r a l , t h a t the number o f k - p l a n e s i n n-space meeting h = (k+1)(n-k) g e n e r a l ( n - k - 1 ) - p l a n e s i s h!k!(k-1)!.•.3!2! , n!(n-1)!...(n-k)! and our two examples a r e s p e c i a l cases of t h i s ,  - 60 Chapter IV MORE RECENT DEVELOPMENTS §1 The Hasse Diagram An o b j e c t t h a t c o n t a i n s a s u r p r i s i n g amount o f i n f o r m a t i o n about t h e Grassmannian i s a c e r t a i n l a t t i c e c a l l e d the Hasse diagram.  I t can be  d e f i n e d as the l a t t i c e a s s o c i a t e d t o the s e t o f Schubert v a r i e t i e s p a r t i a l l y o r d e r e d by i n c l u s i o n . More p r e c i s e l y l e t  (a^,..,a^) be a Schubert sub-  v a r i e t y o f G. (E ) then the Schubert symbol ( a , . , , , a . ) d e f i n e s an i n t e g e r 1  K.  p o i n t i n ]R .  X  XI  Define  ( a , . . . , a ) A ( b , . . . , b ) = (min 1  K.  k  1  k  [a^b  ],  ( a ^ , .. . . a ^ . v ( b . . . , b ) = (max [ a ^ b . ^ ] , 1 >  min [a^.b  ]) ,  min [ a ^ j b ^ ] ) .  k  I t i s easy t o check t h a t A and v form the g r e a t e s t lower bound and l e a s t upper bound r e s p e c t i v e l y , making H, = {(a..,...,a );ft(a ,...,a ) i s a Schubert s u b v a r i e t y o f G (E )} K,n x K X K K n 1  into a distributive l a t t i c e .  When d e a l i n g withi.the f i r s t model o f H, , k,n A and v have g e o m e t r i c meaning i n t h a t fi(a ...,a ) 1}  A fl(b , ...,b ) = fi(a ,...,a ) n S2(b . . . ,b )  k  k  id(a ,. .. ,a, ) v H(b J.  K.  k  , ..,,b,) = X  1>  fc  s m a l l e s t Schubert v a r i e t y c o n t a i n i n g  K.  J2(.a , ,, ,a ) 1?  k  u fi(b  1?  , , «,b ) , fc  B e f o r e e x p l o r i n g the s t r u c t u r e and o t h e r models o f H  we draw t h e tC y n  diagrams i n a few l o w d i m e n s i o n a l  cases,  Many a p p e a l i n g p a t t e r n s p r e s e n t  themselves s t r a i g h t away.  We d e a l w i t h  some t h a t have g e o m e t r i c s i g n i f i c a n c e . The homology b a s i s can be p i c k e d o u t  - 62 -  (2, £ , 3 )  3.fc  C 0,0,0)  - 63 easily,  The g e n e r a t o r s f o r R^ (G^ (E^) , ZZ ) a r e t h e p o i n t s o f  ^ l y i n g on  r  k + x ^ = r i n 3R ,  the h y p e r p l a n e x ^ + x ^ +  I n o u r examples t h i s h y p e r p l a n e  i s a h o r i z o n t a l l i n e , so f o r example H.. (G,. (E_) ,2Z ) = 7L 9 ZZ 4 3 5 g e n e r a t o r s a r e (1,1,1) and (0,0,2) o r -R  and the  (G (E^) ,Z ) = ZZ 9 7L 9 ZZ  g e n e r a t o r s ( 1 , 2 , 2 ) , (1,1,3) and ( 0 , 2 , 3 ) , e t c e t e r a .  with  Thus the B e t t i numbers  can a l s o be r e a d e a s i l y i n t h e same way, so f o r example t h e even B e t t i numbers ( B ^ , ^ , • • • >®ig)  o f  G  3^ 6^ E  The odd ones a r e , o f c o u r s e , a l l  a  r  -*->l>2>3,3,3,3,2,l,l  e  respectively.  zero.  I n a l l t h e examples drawn, we see t h a t t h e " t o p h a l f " and t h e "bottom h a l f " o f t h e Hasse diagram have t h e same shape.  T h i s i s t r u e i n g e n e r a l and k  i s a consequence o f P o i n c a r e d u a l i t y .  V i=l L  - < - > i " 2 k  v  The h y p e r p l a n e II i n 1  n  k  i n t e r s e c t s H, e x a c t l y when dini (G, (E ) ) i s even. I n t h i s case II n H k,n C k n k,n i s t h e s e t o f S c h u b e r t symbols t h a t a r e P o i n c a r e s e l f - d u a l . A p o i n t above J  the h y p e r p l a n e i s r e l a t e d t o i t s P o i n c a r e d u a l below and t h i s p a i r o f p o i n t s d e f i n e s a l i n e segment whose m i d - p o i n t i s i n II. We do n o t , however, see any symmetry happening from l e f t t o r i g h t i n general.  I n t h e case H  and H  and i n t h e case o f H  s  and H ,  diagrams a r e t h e r e f l e c t i o n s o f each o t h e r i n a v e r t i c a l a x i s .  the  This  r e f l e c t i o n i s t h e Hasse diagram's i n t e r p r e t a t i o n o f t h e c a n o n i c a l isomorphism G, (E ) •== G , (E ) . We m e n t i o n a t e c h n i c a l p e c u l i a r i t y h e r e ; k n n-k n J  even though  G, (E ) and G , (E ) a r e c a n o n i c a l l y i s o m o r p h i c t h e i r c o r r e s p o n d i n g Hasse k n n-k n 1  diagrams a r e n o t n e c e s s a r i l y t h e same, y e t G , (E ) and G ,(£**), w h i c h a r e ' n-k ' n n-k n 0  J  J  ?  not c a n o n i c a l l y i s o m o r p h i c , have p r e c i s e l y t h e same Hasse diagram, H  depends o n l y on t h e i n t e g e r s k and n, and H K. y Tl  = H iC y Ti  Thus  i f and o n l y i f T1™*JC y Ti  - 64 k = n-k, w i t n e s s H^ ^ and  ^.  The Hasse diagram then d i s t i n g u i s h e s  between the Grassmannian and i t s ' d u a l :but doesn't c a r e /where the v e c t o r space  comes from;  i n p a r t i c u l a r the Hasse diagram w i l l be the same over  any ground f i e l d IF . The diagrams have been drawn so as t o show how H H^ ^ and how H^ ,. and H^  f i t i n t o H^ ^,  of G. (E ) and G . (E ) f i t i n t o H „ k n n-k n m,2m  and H  f i t into  More g e n e r a l l y the Hasse diagrams  where m = max  (k,n-k). '  This f i t i s a  u s e f u l d e v i c e to h e l p understand the Hasse diagram's i n t e r p r e t a t i o n of duality explicitly.  The same argument used i n the l a t t e r p a r t of example  (2.1.4) shows t h a t fi(0, . .. ,0,a.., . .. ,a, ) o f G. , , (E ,,) i s i s o m o r p h i c to 1 k k+d n+d f!(a, a.) of G, (E ) where d i s the number of zeroes . i n the f i r s t Schubert 1 k K. n symbol, T h i s i n d u c e s an i n j e c t i o n of s e t s  H  k,n ^  \ d,n d +  +  f  °  r  3 1 1  1  ^  k  ^ > d  via (4.1.1)  (a ...,a ) l 5  k  which p r e s e r v e s b o t h bounds.  ( 0 , . . .,0,a ,...,a )  I f we assume k < n-k then H  d i f f e r e n t way i n t o a h i g h e r d i m e n s i o n a l diagram.  H  , —>• H . , n-k,n n-k,n+d  for a l l  , embeds i n a n-k,n  In fact  1 < k < n, '  1 < d'  via ( a ^ , ,, ,, a^)•—>• ( a ^ , , , , , a^) which i s i n d u c e d from c o n s i d e r i n g an ( n - k ) ^ p l a n e Y i n E c E ,, to be an * n n+d ( n - k ) - p l a n e i n -E ^k+d n+d  =  ^n-k n+d  I f we now choose d t o be the i n t e g e r such t h a t 1 , e  *  d  =  n  *"2k we f i n d t h a t the Hasse diagrams of the  -  65  -  Grassmannian and i t s d u a l a r e d i s t i n c t s u b l a t t i c e s o f H The diagrams H\ ^  , , , n-k,2(n-k) 0  l N  a r e p r e c i s e l y those w h i c h do d i s p l a y t h e " s i d e w a y s "  symmetry and we c a l l them the s e l f - r e f l e x i v e Hasse diagrams,  Our d e s i r e i s  to f i n d an e x p l i c i t map A  : H  n - k , 2 ( n - k ) "* n - k , 2 ( n - k ) H  which i s a l a t t i c e isomorphism, by which we mean a b i j e c t i o n o f s e t s p r e s e r v i n g b o t h bounds.  We w i s h i t t o have the p r o p e r t y  H K , n i s H n — l c , n, and t h a t i t have o r d e r two. and i f  CT  eH  t h a t t h e image o f  Such a map we c a l l a r e f l e c t i o n ,  „, , then we c a l l t h e image c the r e f l e c t i o n o f o. n-k,2(n-k)  To  a c c o m p l i s h t h i s l e t r = n-k and N = 2(n-k) = 2 r , choose a f l a g T£ = (EQ  C  E  2  c  •••  c  anc  * from i t d e f i n e a f l a g 3D i n E ^ v i a D. = E° . i  N-i  where V° denotes t h e a n n i h i l a t o r o f V.  L e t X be a p o i n t i n ai(a, ,...,a ) 1 r N o and c o n s i d e r i t s i n t e r s e c t i o n sequence { a ^ } ^ and l e t 3 ^ = dim (X j ) n  We have  a  ±  = dim (X n E )  thus  N-  = dim ( [ X n E ^ ) 0  a ±  = dim (X° + E?) l  = dim (X°) + dim (E?) - dim (X° n E?) = r + N - i - dim (X° n E°) therefore  D  - 66  I  dim  (X°n  ) - a  + (r-i)  that i s  S  o r , on r e p l a c i n g  i by  N-i  ^  i  a  ^"^  +  N-i  (.4.1.2)  0.  = a  . - (r-i) .  M  Thus the i n t e r s e c t i o n sequence of X° w i t h r e s p e c t t o JD the i n t e r s e c t i o n sequence of X w i t h r e s p e c t to E . of {g.} ? _ l 1=0 1  by  (b..,...,b ) and 1 r  i  = (b ,...,b ) •  r  The  f a c t t h a t the r e f l e c t i o n has  and  thus the f a c t t h a t i t i s a b i j e c t i o n of s e t s .  and  following  i n t u i t i o n and  indices.  Denote the l a g sequence  define  A(a ,..;,a )  p r e s e r v e d can be proved by  can be computed from  o r d e r two  i s c l e a r from the  construction  That the bounds  are  t a k i n g a c l o s e l o o k at the i n t e r s e c t i o n sequences  The  d e t a i l s p r o v i d e l i t t l e i n the way  of g e o m e t r i c  are o m i t t e d h e r e .  G, (E ) embeds i n G (E ) as the Schubert v a r i e t y where the number of zeroes i s n-2k.  We  ft(0,...,0,n-k,...,n-k)  compute i t s r e f l e c t i o n .  X £ a) (0, . .. , 0, n-k,. . ., n-k)  i ( X ) = (0,1,2,...,n-2k, n-2k,...,n-2k, n-2k+l,n-2k+2,,,.,n-k)  > " V  \ I,  (n-2k) terms w i t h r e s p e c t to IE  and  •  -y—•  (n-k)  a c c o r d i n g to  '  t  terms (4,1,2)  t  k terms  i-  Let  - 67 with respect  to B ,  -  Thus X ° e w ( k , k , , , , , k ) .  Thus we have shown t h a t  r e f l e c t i o n of ( 0 , , . , , 0 , n - k , , , , , n - k ) , which r e p r e s e n t s which r e p r e s e n t s  , (E" ) , n-k n an isomorphism between H, k,n The  G  r e f l e c t i o n map  G (E ), i s (k, , ,,,k)  I t f o l l o w s t h a t the r e f l e c t i o n map and H  the  restricts  to  , n-k,n  r e c o n c i l e s the two  choices  i n the l i t e r a t u r e of  the  b a s i s f o r the cohomology r i n g ( o r , e q u i v a l e n t l y the homology c o n s i d e r e d a r i n g w i t h the i n t e r s e c t i o n p r o d u c t ) .  Our  c h o i c e i s t h a t of  as  Griffiths,  whereas o t h e r s , n o t a b l y K l e i m a n and L a k s o v , choose the s p e c i a l Schubert c y c l e s to be of the form (j,n-k,n-k,.,.,n-k)  for  1 < j <  So, where we have k s p e c i a l Schubert c y c l e s on counts (n-k)  s p e c i a l Schubert c y c l e s .  n-k.  (G, (E ) the o t h e r K. n  choice  A g a i n by l o o k i n g a t the i n t e r s e c t i o n  sequences i t i s easy to see t h a t these (n-k)  a l t e r n a t e Schubert c y c l e s  s i m p l y the r e f l e c t i o n s of our c h o i c e of (n-k) G . fe ) . n-k n  s p e c i a l Schubert c y c l e s  One  would hope t h a t G i a m b e l l i ' s f o r m a u l a i s c o m p a t i b l e w i t h  reflection.  are on  the  S i n c e c a l c u l a t i o n s i n h i g h e r d i m e n s i o n become v e r y cumbersome  v e r y q u i c k l y we  show the t r u t h of t h i s o n l y i n the example of G^iE^).  c h o i c e of s p e c i a l Schubert c y c l e s then i s  = (1,2),  the o t h e r c h o i c e i s a-^ = (.1,2),  Now  a  A  =  CT  (0,2).  = ( 1 , 1 ) , whereas  A-1  (,2-ui2-A) = ( y , A ) * =  * A ' u a  a  Vl ' Vl  a  A ' %  y+l . y  s i n c e a t l e a s t one  of A - 1 and y + l  Giambelli's formula.  i s o u t s i d e the range s t i p u l a t e d by  U s i n g the r e f l e c t i o n s i n s t e a d we  Our  have  -  Note t h a t codim a.  codim o. f o r i  X  12  1  x  o  68  a  • a  A  ?  • A = a '•- a A ..y y  a  y  and so by F i e r i ' s  • a  A =  formula.  a  I n g e n e r a l P i e r i ' s f o r m u l a i s p r e s e r v e d under r e f l e c t i o n s i n c e the o n l y p r o p e r t y of a s p e c i a l Schubert c y c l e i t uses i s i t s codimension,  which i s  preserved, R e t u r n i n g t o the Hasse diagram we c a l l a Schubert symbol a and immediate predecessor  of 3 i n case a < 3 and t h e r e does not e x i s t y such t h a t a < y < &  where < i s the t o t a l o r d e r i n g on the Hasse diagram. immediate p r e d e c e s s o r  Geometrically  an  o f a Schubert v a r i e t y ft i s a Schubert s u b v a r i e t y of  codimension 1 i n ft. P r o p o s i t i o n (3.2.1) can be r e s t a t e d then as uj(.a  , . . . , a, ) = ft (a , ...,a ) \ a l l immediate  predecessors  of ft (a , . . . ,a, ) . S i n c e a)(a^,...,a^)  i s smooth, the s i n g u l a r l o c u s of  i n the u n i o n of i t s p r e d e c e s s o r s . We l e t us l o o k a t the example of  ft(l,2).  ft(a^,...,a ) k  i s contained  d e s c r i b e p r e c i s e l y w h i c h ones, but In H  . the symbol (1,2)  first  i s the o n l y  one w i t h more than one immediate p r e d e c e s s o r , moreover, ft(1,2) i s the o n l y s i n g u l a r Schubert v a r i e t y in•-G ( E ^ ) . In 1974,  T h i s i s not a c o i n c i d e n c e .  Svanes p u b l i s h e d a paper e n t i t l e d Coherent Cohomology on  Schubert Subschemes of F l a g Schemes and A p p l i c a t i o n s i n which he  constructed  an e x p l i c i t r e s o l u t i o n of the s i n g u l a r l o c u s of an a r b i t r a r y Schubert v a r i e t y over an a r b i t r a r y ground f i e l d ,  The  complete p r o o f o f t h i s i s h i g h l y  t e c h n i c a l and beyond the scope of t h i s d i s c u s s i o n , we o n l y quote the r e s u l t ,  - 69 Theorem (.4.1.3) Let  k  Q  be zero and a^ > 1,  a  k. . l-l  <  I f k^. i s d e f i n e d i n d u c t i v e l y by  \ . • +1 " k . ,+2 i - i i - i a  =  =  a  k.  s-1 then S i n g (£2 (a , ., ., a )) = M £2. j=l 1  k  £1 =fl(a , . . . , a ^  \.+l l  I  where k  = k  J  ,a 3-1  <  -l,a - l , . . . ,a - l , a 3-1 3 3 2  3-1  a  2  ,...,a) .  3  Note t h a t the h y p o t h e s i s a^ & 1 i s s i m p l y a c o n v e n i e n c e s i n c e  To i l l u s t r a t e t h i s we c o n s i d e r an example l a r g e enough to see what i s happening, say S i n g £2(1, 1,1,2,3p,4,4,4,5) = £2(0,0,0,0,3,3,4,4,4,5) 0.(1,1,1,1,1,3,4,4,4,5) £2(1,1,1,2,2,2,2,4,4,5) £2(1,1,1,2,3,3,3,3,3,3) . Note t h a t (2.2,2) i s a l s o a s p e c i a l case o f (4.1.3) i . e . S i n g £2(1,2) = £2(0,0) .  A consequence o f t h i s can be seen on the Hasse d i a g r a m , namely Corollary  (4.1.4)  A Schubert v a r i e t y i s s i n g u l a r i f and o n l y i f the c o r r e s p o n d i n g symbol i n the Hasse diagram has a t l e a s t two immediate p r e d e c e s s o r s ,  - 70 Proof: Suppose ( a , , , n a , l has a t l e a s t two immediate p r e d e c e s s o r s then t h e r e (  e x i s t 1 < 1 < j < k such t h a t  a. T < a. = a . . < a., i-l i J-l J Consequently ft C a ,  a. . ,a .-1, . , . ,a . , - l , a . 5.1-1' i  1  j - l  -l,a.,„,.,, ,a,)  j - l  i s c o n t a i n e d i n the s i n g u l a r l o c u s of ti(a^, , ,, ,a^)  J+2  k  and i s non-empty.  C o n v e r s e l y suppose (a..,..,,a,) has o n l y one p r e d e c e s s o r .  L e t a. be t h e  JC  X  f i r s t non-zero i n t e g e r i n (a^,...,a^) CO,...,0,a - l , a  X  then the unique immediate  predecessor i s  ,...,a ). k  We c l a i m t h a t a. = a. f o r a l l i > i , f o r i f n o t l e t a. be the f i r s t i J i i n the Schubert symbol  k  fi(a^,...,a^)  i s of the form ft(0, . . . ,0,a,. . . ,a) which, bby  i s the Grassmannian In  5  ( 0 , . . . ,0,a .,...,a, ) such t h a t a. > a. then J k i J  (.0, . . . ,0,a., . . .,a ,a - 1 , .. .,a ) would be a d i f f e r e n t immediate So  integer  G..(E  ,,) and i s t h e r e f o r e smooth,  predecessor.  example ( 2 . 1 . 4 ) ,  q.e.d.  t h e c o u r s e o f t h e p r o o f we have e s t a b l i s h e d  Corollary  (4.1.5)  A Schubert v a r i e t y i s smooth i f and o n l y i f i t i s a  Grassmannian.  I t i s now easy t o count the number of smooth Schubert v a r i e t i e s i n G,(E  ).  There a r e Q, (0, . . . ,0) and each ft(0, . .. ,0,a, . . . ,a) where  l a s t i p l a c e s f o r a l l 1 < i < k and runs from 1 t o ( n - k ) . k ( n - k ) + l smooth Schubert v a r i e t i e s , the  o c c u p i e s the  T h i s makes  T h i s a l s o g i v e s us the c u r i o u s f a c t  that  number o f s i n g u l a r S c h u b e r t v a r i e t i e s i s e x a c t l y t h e c o d i m e n s i o n o f  G ( E ) as a p r o j e c t i v e v a r i e t y v i a the P l u c k e r embedding, k  a  n  A t the time of  w r i t i n g we see no i n t r i n s i c g e o m e t r i c r e a s o n f o r t h i s though we b e l i e v e t h e r e is  one,  - 71 The P l u c k e r c o o r d i n a t e s themselves, however, form a n o t h e r model o f the t h e v e r t e x (a , , ,,a.) as JX 1 K ^  Hasse diagram by l a b e l i n g  ,  r  l''''?  3  and t h i s  ]j  does have g e o m e t r i c c o n t e n t by v i r t u e of P r o p o s i t i o n (3,2,2) i , e , a p o i n t of ( a , , , , , a ) w  1  k  i s a p o i n t of n (a , ,' , ,a ) where x^ f  fc  +  a +k ^  1  0  '  ' ' ' '' k  1  I t i s p o s s i b l e t o show t h a t , f o r a s e l f - r e f l e x i v e Hasse diagram H  ,  P o i n c a r e d u a l i t y commutes w i t h r e f l e c t i o n , i . e . t h e symbol A  (a^,.,.» ) a  *  k  i s unambiguous.  Suppose 1<  X <... < A, ^ 2k and suppose I  1 < y ^ < ... < (y^,...,y ) k  order. x  < 2k  &  i s t h e complementary sequence, by which we mean t h a t  i s the set  { l , 2 , . . . , n } \ {A^,....X^}  Now suppose t h a t (a ,...,a ) i s t h e v e r t e x of HX  A^. ••..,A  1  K-  A  (a ,...,a ) = 1  k  =  V  p=l  y^,...,y  We c o n j e c t u r e t h a t  * 1  then t h e q u a d r a t i c r e l a t i o n  +  (-1) x Pq  X  s  1  does not c o l l a p s e on W^. n W  p ^,q  k  <"  1  f o r 1 < q < k, thus  to f i n d independent q u a d r a t i c r e l a t i o n s  q p A  k  the c o n j e c t u r e might h e l p  l o c a l l y , o f which t h e r e s h o u l d be  ( ) - k(n-k) - 1 = t h e number o f s i n g u l a r Schubert k  .  k  (b ,...,b ) .  I f I and J a r e complementary sequences  k  corresponding to  R) Z K  iC  and t h a t ( b b , ) c o r r e s p o n d s t o x  k  X X  arranged i n a s c e n d i n g  varieties.  §2  72  -  C o n c l u d i n g Remarks The i n t e r s e c t i o n t h e o r y used h e r e r e l i e d h e a v i l y  were w o r k i n g over t h e complex numbers. applied 1930.  on t h e f a c t . t h a t we  I t was developed by L e f s c h e t z and  t o t h e f o u n d a t i o n s o f t h e Schubert c a l c u l u s by v a n d e r Waerden i n Ehresmann found t h e c e l l d e c o m p o s i t i o n i n 1934 and developed some  g e n e r a l r e s u l t s about c e l l complexes, t h a t a r e now s t a n d a r d , t o prove t h e basis  theorem.  A l l o f t h i s however was t o p o l o g i c a l .  Hodge produced t h e f i r s t p u r e l y a l g e b r a i c and  1942 w i t h t h e papers The base f o r a l g e b r a i c  i n t e r s e c t i o n t h e o r y i n 1941 v a r i e t i e s of a given  d i m e n s i o n en a grassmannian v a r i e t y and T h e - i n t e r s e c t i o n grassmannian v a r i e t y .  He proved t h e b a s i s  an a r b i t r a r y s p e c i a l Schubert c y c l e shown t h e case where 1 = 1 .  formulae f o r a  theorem, and P i e r i ' s f o r m u l a f o r  o^, w h i l e v a n d e r Waerden had f i r s t  Hodge then used P i e r i ' s f o r m u l a t o p r o v e  Giambelli's formula. A g r e a t many i n t e r s e c t i o n t h e o r i e s  were d e v e l o p e d a f t e r Hodge's and  i n t h e f o l l o w i n g we make no p r e t e n s i o n s o f completeness. n o t a b l e was t h e Chow r i n g , d e f i n e d as f o l l o w s : group on a l l i r r e d u c i b l e s u b v a r i e t i e s a l g e b r a i c a l l y closed  field.  Perhaps t h e most  l e t C (V) be t h e f r e e  abelian  o f V, a p r o j e c t i v e v a r i e t y over an  I f X and Y a r e i r r e d u c i b l e s u b v a r i e t i e s  of V  then they a r e s a i d t o i n t e r s e c t p r o p e r l y i f  codcodim:(Z) = codim (X) + codim (Y) f o r e v e r y i r r e d u c i b l e component Z o f X n Y.  The a b s t r a c t  d e f i n i t i o n of  i n t e r s e c t i o n m u l t i p l i c i t y t h a t was mentioned i n t h e i n t r o d u c t i o n was i n v e n t e d by.Se.rre i n 1965 i n a paper e n t i t l e d A l g e b f e l o c a l e - m u l t i p l i c i t e s , and i s d e f i n e d i n terms o f h o m o l o g i c a l a l g e b r a .  We denote t h e m u l t i p l i c i t y o f a  component Z o f X n Y by I(X,Y;Z) and so i t i s p o s s i b l e  to define a product  - 73 X . Y =  I I(X,Y;Z)Z , ZcXnY  whenever X and Y i n t e r s e c t p r o p e r l y . on C (V) t h a t guarantee  One  now  considers various equivalences  c h o i c e s o f r e p r e s e n t a t i v e s f o r each p a i r of  e q u i v a l e n c e c l a s s e s , so t h a t the r e p r e s e n t a t i v e s meet p r o p e r l y .  There i s a  h i e r a r c h y of these e q u i v a l e n c e s and the s t r o n g e s t o n e . i s c a l l e d l i n e a r o r r a t i o n a l equivalence.  Two v a r i e t i e s are l i n e a r l y e q u i v a l e n t i f they a r e b o t h  members o f an a l g e b r a i c system o f s u b v a r i e t i e s p a r a m e t r i z e d by TP  \  The Moving Lemma For any two v a r i e t i e s X,Y-on V t h e r e i s a v a r i e t y Y', t o Y, such t h a t X and Y' i n t e r s e c t p r o p e r l y .  I f X,Y,Z  l i n e a r l y equivalent  a r e v a r i e t i e s such t h a t  X i s l i n e a r l y e q u i v a l e n t t o Y then whenever X • Z and Y • Z a r e d e f i n e d A  X . Z i s l i n e a r l y e q u i v a l e n t t o Y • Z.  The q u o t i e n t of C (X) by  linear.  e q u i v a l e n c e i s the l a r g e s t r i n g f o r w h i c h the i n t e r s e c t i o n p r o d u c t i s d e f i n e d everywhere.  T h i s r i n g i s c a l l e d the Chow r i n g .  L i n e a r e q u i v a l e n c e can be weakened. f a m i l y p a r a m e t r i z e d by v a r i e t y U.  I n s t e a d o f h a v i n g the  continuous  we have i t p a r a m e t r i z e d by a q u a s i - p r o j e c t i v e  The r e s u l t i n g r e l a t i o n i s c a l l e d a l g e b r a i c e q u i v a l e n c e .  The  h i e r a r c h y i s as f o l l o w s : l i n e a r equivalence=>algebraic  equivalence=>homological  equivalence.  Here h o m o l o g i c a l e q u i v a l e n c e means membership i n the same W e i l cohomology c l a s s , where a W e i l cohomology i s an i n v a r i a n t on v a r i e t i e s over a c h a r a c t e r i s t i c z e r o f i e l d t h a t behaves f o r m a l l y l i k e s i n g u l a r cohomology on manifolds. There i s even weaker e q u i v a l e n c e define'.d as f o l l o w s ;  consider  - 74 C*(V) + C (.V) + 2Z n  n ' * where C (V) i s t h e subgroup of C (V) generated by s u b v a r i e t i e s . o f ' c o d i m e n s i o n  * n ( i . e . points)v, t h e f i r s t map i s t h e p r o j e c t i o n (C (V) i s a graded  group),  and t h e second map adds up t h e c o e f f i c i e n t s o f t h e l i n e a r c o m b i n a t i o n s :  The  second map i s c a l l e d t h e augmentation.' I f X • Y i s d e f i n e d , then i t s image under t h e composite map i s an i n t e g e r c a l l e d t h e i n t e r s e c t i o n number of X and Y.  I t i g n o r e s components o f X n Y o f p o s i t i v e dimensions  only points with t h e i r m u l t i p l i c i t i e s .  counting  If  <X,z) = <Y,z) <t'6v: e v e r y Z f o r w h i c h t h e p r o d u c t i s d e f i n e d then X and Y a r e s a i d t o be numerically equivalent.  By analogy w i t h s e c t i o n 1 o f c h a p t e r I I I one would  expect homological equivalence =>numerical equivalence and i n d e e d i t does, b u t t h e o p p o s i t e i m p l i c a t i o n i s more i n t e r e s t i n g . Grothendieck's  1958 p u b l i c a t i o n Sur quelques p r o p r i e t e s fondamentale en  theor'jefetes i n t e r s e c t i o n s showed t h a t t h e r e i s a c e r t a i n g e n e r a l c l a s s o f v a r i e t i e s , w h i c h c o n t a i n s Grassmanmians and f l a g v a r i e t i e s , w i t h t h e p r o p e r t y that  numerical equivalence =>homological equivalence. He a c t u a l l y proved a more g e n e r a l r e s u l t b u t i t i s n o t needed h e r e . Laksov,  i n 1972, c o n s t r u c t e d an i n t e r s e c t i o n t h e o r y over an a r b i t r a r y  ground f i e l d i n h i s paper e n t i t l e d A l g e b r a i c C y c l e s on Grassmannian V a r i e t i e s . He p r o v e d , u s i n g t h i s t h e o r y , t h e b a s i s theorem and v e r s i o n s o f b o t h  Pieri's  - 75 and G i a m b e l l i ' s f o r m u l a e .  I n c o n t r a s t w i t h Hodge's-method, Leksov proved  G i a m b e l l i ' s f o r m u l a f i r s t and used i t t o p r o v e P i e r i ' s Using Lefschetz' i n t e r s e c t i o n  formula.  t h e o r y we s o l v e d two examples i n  enumerative geometry where t h e f i g u r e s were l i n e a r s p a c e s . t h a t t h i s c a n always be done i n c h a r a c t e r i s t i c  K l e i m a n has shown  z e r o , and more g e n e r a l l y f o r  any f i g u r e s where t h e g e n e r a l l i n e a r group a c t s t r a n s i t i v e l y on t h e parameter variety.  T h i s i s n o t t h e case f o r c o n i e s ;  t h e g e n e r a l l i n e a r group i n t h i s  case has f o u r o r b i t s , namely t h e s e t o f n o n - s i n g u l a r c o n i e s , t h e s e t o f p a i r s o f d i s t i n c t l i n e s , t h e s e t o f double l i n e s w i t h d i s t i n c t f o c i and t h e s e t o f double l i n e s w i t h double f o c i .  In certain  cases however i t i s p o s s i b l e t o  s o l v e an enumerative problem i f t h e f i g u r e s a r e n o t " s t r a i g h t . "  F o r example,  K l e i m a n and Laksov prove t h a t t h e number ( o r i g i n a l l y found by S c h u b e r t ) o f 4  l i n e s common t o two q u a d r i c h y p e r s u r f a c e s  i n ]P i s 16. A l s o , i n t h e i r 3  a r t i c l e Schubert C a l c u l u s , they show t h a t t h e number o f l i n e s i n IP which simultaneously intersect  f o u r g i v e n c u r v e s C^, C^, C^, C^, i f f i n i t e , i s e q u a l  to 26 6 6 6 , 1  2  2  4  where 6^ i s t h e degree o f C^ and t h e number i s counted w i t h m u l t i p l i c i t i e s . The program g i v e n by H i l b e r t ' s much c l a s s i c a l work s t i l l t o v e r i f y .  f i f t e e n t h problem i s immense, t h e r e i s Some a s p e c t s o f t h e problem have been  s o l v e d r e p e a t e d l y , b u t , i n t r u t h , we must s t i l l c o n s i d e r H i l b e r t ' s problem u n s o l v e d .  fifteenth  - 76 BIBLIOGRAPHY 1.  B o r e l , A., L i n e a r A l g e b r a i c Groups, W.A. Benjamin I n c . , 1969.  2.  Chern, S.S., Complex M a n i f o l d s Without P o t e n t i a l Theory, D. Van N o s t r a n d Co. I n c . , 1967.  3.  D o l d , A., L e c t u r e s on A l g e b r a i c Topology, S p r i n g e r - V e r l a g ,  4.  Ehresmann, C , "Sur l a t o p o l o g i e de c e r t a i n s espaces homogenes", Ann. Math.  [35(1934), 396-443].  5.  F u l t o n , W., A l g e b r a i c Curves, W.A. Benjamin I n c . , 1969.  6.  G i a m b e l l i , G.Z.,  " S u l p r i n c i p i o d e l l a c o n s e r v a t i o n e d e l numero",  J a h r e s b . d e u t s c h . Math.-Ver. 7.  1972.  [13(1904), 545-556].  Greenburg, M., L e c t u r e s on A l g e b r a i c Topology, W,A. Benjamin I n c . , 1967.  8.  G r i f f i t h s , P. and Adams, J . , T o p i c s i n A l g e b r a i c and A n a l y t i c Princeton University Press,  9.  10.  1974.  G r i f f i t h s , P. and H a r r i s , J . , P r i n c i p l e s o f A l g e b r a i c Geometry, W i l e y and Sons,  Geometry,  John  1978.  G r o t h e d i e c k , A., "Sur quelques p r o p r i e t e s fondamentales en t h e o r i e des i n t e r s e c t i o n s " , S e m i n a i r e C. C h e v a l l e y E.N.S. (1958).  11.  H a r t s h o r n e , R., Rees, E. and Thomas, E., "Non-smoothing  of algebraic  c y c l e s on Grassmann v a r i e t i e s " , B u l l e t i n o f the American M a t h e m a t i c a l S o c i e t y [80, 5 ( 1 9 7 4 ) , 847-851]. 12.  Hodge, W.V.D., "The base f o r a l g e b r a i c v a r i e t i e s o f g i v e n d i m e n s i o n on a grassmannian v a r i e t y " , J o u r n a l Lond. Math. Soc. [16(1941), 245-255].  13.  Hodge, W.V.D., "The i n t e r s e c t i o n formulae f o r a grassmannian v a r i e t y " , J o u r n a l Lond. Math. Soc. [17(1942), 48-64].  14.  Hodge, W.V.D. and Pedoe, D., Methods o f A l g e b r a i c Geometry, U n i v e r s i t y P r e s s , v o l . I I (1952), r e p r i n t e d  1968.  Cambridge  - 77 15.  K l e i m a n , S.L., " A l g e b r a i c c y c l e s and t h e W e i l c o n j e c t u r e s " , D i x exposes s u r l a cohomologie des schemes, N o r t h H o l l a n d (1968).  16.  K l e i m a n , S.L., "The t r a n s v e r s a l i t y o f a g e n e r a l t r a n s l a t e " , C o m p o s i t i o Math. [28, 3(1974),  17.  287-297].  K l e i m a n , S.L. "Problem 15. R i g o r o u s F o u n d a t i o n o f Schubert's  Enumerative  C a l c u l u s " , A.M.S. P r o c e e d i n g s o f Symposia i n Pure Mathematics  [28(1976),  445-482]. 18.  K l e i m a n , S.L. and Laksov, D., "Schubert C a l c u l u s " , Am. Math. Monthly [79(1972),  19.  1061-1082].  Lakson, D., " A l g e b r a i c c y c l e s on Grassmann v a r i e t i e s " , Advances i n Math. [9(1972),  267-295].  20.  L e f s c h e t z , S. A l g e b r a i c Topology, American M a t h e m a t i c a l S o c i e t y , 1942.  21.  M i l n o r , J . and S t a s h e f f , J . , C h a r a c t e r i s t i c C l a s s e s , P r i n c e t o n U n i v e r s i t y P r e s s , 1974.  22.  Mumford, D., A l g e b r a i c Geometry I , Complex P r o j e c t i v e V a r i e t i e s , S p r i n g e r - V e r l a g , 1976.  23.  P o n c e l e t , J.V., T r a i t e des p r o p r i e t e s p r o j e c t i v e s des f i g u r e s , G a u t h i e r - V i l l a r s , P a r i s (1822), second e d i t i o n , P a r i s (1865).  24.  P o r t e o u s , I.R., "Simple s i n g u l a r i t i e s o f maps", L i v e r p o o l S i n g u l a r i t i e s Symposium I , L e c t u r e notes i n math., S p r i n g e r - V e r l a g [192(1971), 286-307] .  25.  S c h u b e r t , H.C.H., K a l k u l d e r abzahlenden Geometrie, Teubner, L i e p z i g (1879) .  26.  S e r r e , J.P., A l g e b r e l o c a l e - m u l t i p l i c i t i e s " , L e c t u r e n o t e s i n math., 11, S p r i n g e r - V e r l a g (1965).  27.  S e v e r i , F., " S u l p r i n c i p i o d e l l a c o n s e r v a z i o n e d e l numero", R e n d i c o n t i d e l C i r c o l o Matematico  d i Palermo  [33(1912),  313-327].  - 78 28.  Severi,  F., " S u i fondamenti d e l l a g e o m e t r i a numerative e s u l l a  t e o r i a d e l l e c a r a t t e r i s t i c h e " , A l t i d e l R. I n s t i t u t o  29.  [75(1916),  1121-1162].  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