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The Schubert calculus Higham, David Paul 1979

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THE SCHUBERT CALCULUS by DAVID PAUL HIGHAM B . S c , 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 i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1979 David Paul Higham, 1979 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Mathematics The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 D a t e October 1, 1979 i i ABSTRACT An enumerative problem asks the f o l l o w i n g type of question; how many f i g u r e s ( l i n e s , planes, conies, c u b i c s , etc.) meet t r a n s v e r s e l y (or are tangent to) a c e r t a i n number of other f i g u r e s i n general p o s i t i o n ? The l a s t century 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 Schubert, 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 that when c o n d i t i o n s of an enumerative problem are v a r i e d continuously then the number of s o l u t i o n s i n the general 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 . Schubert c a l l e d i t the p r i n c i p l e of conservation of number. To date 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 only i s o l a t e d cases have been solved where the f i g u r e s are curved. H i l b e r t considered the Schubert c a l c u l u s of s u f f i c i e n t importance to request 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 tra c e the f i r s t foundation of the c a l c u l u s due p r i m a r i l y to Lef s c h e t z , van der Waerden and Ehresmann. 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 , being a summary of Kleiman's ex p o s i t o r y 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 problem. We describe the Grassmannian and i t s Schubert s u b v a r i e t i e s more fo r m a l l y and describe e x p l i c i t l y the homology of the Grassmannian which gives a foundation f o r the 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 . F i n a l l y we compute two examples and b r i e f l y mention some more recent developments. i i i TABLE OF CONTENTS ABSTRACT . i i TABLE OF CONTENTS i i i LIST OF FIGURES , , . . . i v ACKNOWLEDGEMENT v In t r o d u c t i o n 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 Man i f o l d 22 §4 The U n i v e r s a l Bundle over the 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 Ring S t r u c t u r e i n Homology 51 Chapter IV MORE RECENT DEVELOPMENTS §1 The Hasse Diagram 60 §2 Concluding Remarks 72 BIBLIOGRAPHY 76 i v LIST OF FIGURES Figure 1 ^2 3 6 1 Figure 2 ^ 6 1 Figure 3 ^2 h 6 1 Figure 4 H 3 5 ^ Figure 5 R2 5 6 1 Figure 6 H Q a 62 V ACKNOWLEDGEMENT I am indebted mainly to Lar r y Roberts, my a d v i s o r , f o r h i s patience i n seeing t h i s work through to i t s com-p l e t i o n . I would a l s o l i k e to thank Jim C a r r e l l f o r the o r i g i n a l idea and f o r many of the u s e f u l references. G r a t i t u d e i s a l s o due to Roy Douglas, Mark Goresky, Jim Lewis, Ron R i d d e l l and B i l l Symes f o r t h e i r sympathetic and sometimes i n s p i r i n g d i s c u s s i o n s . Worthy of mention a l s o are those who helped 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 support, tea and sympathy. Those uppermost i n my mind are Roy Douglas, Ed G r a n i r e r , Fred Henry, Mike Margolick, Ken S t r a i t o n and Scott Sudbeck. F i n a l l y , a word of thanks to the "behind-the-scenes" people Mrs. MacDonald and Kathy Agnew f o r t h e i r many kindnesses and bu 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 her i n t e l l i g e n t t y p i n g of the manuscript. - 1 -INTRODUCTION During the l a s t century work i n geometry was h i g h l y i n t u i t i v e . This was e s p e c i a l l y true of the so c a l l e d enumerative geometry, which attempted to answer the question "How many f i g u r e s i n general s a t i s f y a p r e s c r i b e d set of geometric c o n d i t i o n s ? " A simple example of t h i s i s to f i n d the number of l i n e s that meet four given l i n e s i n general p o s i t i o n i n 3-space. Poncelet began work on questions of t h i s nature w h i l e i n a Russian m i l i t a r y p r i s o n at Saratow i n 1813. He published a paper e n t i t l e d T r a i t e  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 i n which he introduced a 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 . Roughly put, the p r i n c i p l e s t a t e s that the number of s o l u t i o n s to an enumerative problem does not change i f the parameters are v a r i e d continuously. The p r i n c i p l e was not properly 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 before the paper was even published. In s p i t e of Cauchy's i n f l u e n c e , which created some p r e j u d i c e , the p r i n c i p l e obtained widespread p o p u l a r i t y and the r e s u l t i n g controversy has not been completely r e s o l v e d even to t h i s very day. Hermann Casar Hannibal Schubert was a p r o l i f i c geometer and, having r e v i v e d the p r i n c i p l e , used i t to c a l c u l a t e the s o l u t i o n s to an astounding number of enumerative problems. His f e r t i l e mind produced numbers that were of t e n i n the tens and hundreds of thousands or more, long before the advent of the 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 the development of the i l l - f a t e d " d i f f e r e n c e engine" of Charles Babbage. In 1874 Schubert 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 to avoid the p r e j u d i c e . Schubert, however, was not s a t i s f i e d that t h i s name embodied the n o t i o n of continuous v a r i a t i o n , and so the p r i n c i p l e received i t s f i n a l baptism, two years l a t e r , as the p r i n c i p l e of conservation of number. Despite the wealth of h i s - 2 -c o n t r i b u t i o n s to enumerative geometry though, Schubert r e a l i z e d that the p r i n c i p l e s t i l l needed to be confirmed. Returning now to the example mentioned above we w i l l see how Schubert answered the question. Let the four l i n e s be L^, L^, L^ and L^ and assume that they are i n general p o s i t i o n . Now move L^ so that i t i n t e r s e c t s L^ at P , and move L^ so that i t i n t e r s e c t s L^ at Q . The l i n e s are now i n " s p e c i a l p o s i t i o n " and i t i s easy to count the l i n e s that pass through a l l these four l i n e s . One l i n e , L, i s defined 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 plane, the l i n e of i n t e r s e c t i o n L', of the two planes i s a second l i n e passing through a l l four l i n e s i n s p e c i a l p o s i t i o n . Suppose there i s a t h i r d l i n e L". To avoid n o t a t i o n a l clumsiness we w i l l denote the s p e c i a l i z e d l i n e s by L^ and L^ a l s o . Now l e t R. be.the p o i n t of i n t e r s e c t i o n of L" and L.. Since. L" i s d i s t i n c t 1 1 from L, L" does not pass through both P and Q . Thus i f L" passes through P , say, then as Q i s the only point common to L^ and L^, R^ and R^ are d i s t i n c t , i . e . contains at l e a s t three p o i n t s . F i r s t suppose that A = { P , R ^ , R ^ } (or e q u i v a l e n t l y { Q J R ^ J R ^ } ) . The l i n e defined by R ^ and R ^ ( i . e . L" i t s e l f ) l i e s i n the plane spanned by L^ and L^, f o r c i n g P to l i e i n that plane a l s o . But then the two planes must be c o i n c i d e n t , i n which case there would be an i n f i n i t e number of l i n e s passing through the fou r given l i n e s . Secondly suppose that the R ^ are a l l d i s t i n c t ; then L" l i e s i n both planes, and since L' and L" are d i s t i n c t t h i s again f o r c e s the two planes to be co i n c i d e n t . - 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 is f i n i t e in the f i r s t place. Incidentally, another degenerate case to avoid is the possibility 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 in 1879 and in i t he computed number after number of solutions to enumerative problems. A l l the examples calculated,. like 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 validity has s t i l l not been established. Schubert later worked in higher dimensions. In 1886.Schubert obtained the number h!k!(k-1)!...3!2! n!(n-l)!...(n-k)! of k-planes in 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 is best expressed :in the statement of Hilbert's fifteenth problem, the text of which, translated in 1902 by Newson i s as follows: The problem consists in this: To establish rigorously and with an exact determination of the limits of their validity 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 -Although the algebra of today guarantees, 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 processes of e l i m i n a t i o n , yet f o r the proof 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 process of e l i m i n a t i o n i n the case of equations of s p e c i a l form i n such a way that the degree of the f i n a l equations 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 foreseen. Poncelet had, i n 1822, claimed, that the p r i n c i p l e could be v e r i f i e d a l g e b r a i c a l l y but didn't do so because he f e l t that the problem should be viewed purely g e o m e t r i c a l l y . Schubert f e l t the same way, though he s t a t e d i n h i s book that 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 to saying that the number of roots of an equation doesn't change i f 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, de Jonquieres had t r i e d to e s t a b l i s h t h i s by applying the fundamental theorem of algebra. But s i n c e a polynomial can have repeated roots we might expect to have m u l t i p l i c i t i e s to contend with sometimes., and indeed such i s the case as, r e t u r n i n g to our example, we now 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 that the plane of L^ , and i s p a r a l l e l but d i s t i n c t from the plane of and L^, then r a t h e r than say there i s no l i n e passing through a l l four l i n e s , we a l l o w the s o l u t i o n at i n f i n i t y . 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 imaginary p o i n t s , and so the ambient space i s complex p r o j e c t i v e 3-space. We 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 cannot r u l e out the p o s s i b i l i t y of only one s o l u t i o n as the f o l l o w i n g w i l l show. Choose L^, and to be three skew l i n e s and choose P^eL^. Let TI^  be the span of P^ and L^ and 11^ be the span of P^ and L^. Since L^ and 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 d i s t i n c t . Now choose L, to be n..n:ll-i l e t P. = L. n L. for i = 1,2,3 and these points 4 2 3 i 4 1 ' are d i s t i n c t since the three l i n e s are skew. L, i s a l i n e that meets a l l 4 f o u r . l i n e s L ^ , so now assume that L i s a d i f f e r e n t l i n e passing through a l l L_^ . Since i s determined by P. and P^ L cannot pass through both of these points, so assume, without loss of generality, that L does not pass through 7^. L meets both and so l i e s i n the plane defined by them, but since L. and L. both l i e i n n~ t h i s plane i s exactly n„• Thus we have 2 4 2 2 (ji ^ L n '£112^1"^ = {P-j} i . e . L meets at P^. S i m i l a r l y L meets at P^. But then L i s defined by P^ and P^ and so must be i t s e l f , contra-d i c t i n g the hypothesis that L and are d i s t i n c t . Thus i s the unique l i n e meeting a l l four given l i n e s . The p r i n c i p l e of conservation of number can s t i l l be salvaged so long 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 is may seem somewhat contrived u n t i l we remember that t h i s configuration of l i n e s i s a c t u a l l y a degenerate case where, the two solutions of the general case have coalesced into one of the four l i n e s , namely L^. The p r i n c i p l e s t a r t s to become c l a r i f i e d when we state i t l i k e t h i s : i f the number of solutions to an enumerative problem i s f i n i t e then that number, counted with 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 solutions i n the general case. The problem of counting m u l t i p l i c i t i e s i s decidedly d i f f i c u l t , though i t was approached with great courage, and m u l t i p l i c i t i e s were assigned with great a l a c r i t y by the adept c l a s s i c a l geometers. This problem i s c e n t r a l to the rigorous foundation of Schubert's enumerative calculus, i n f a c t , i n t h e i r a r t i c l e GEometrie Enumerative of 1915, Zeuthen and P i e r i consider i t of such fundamental s i g n i f i c a n c e that they state that obtaining i t s s o l u t i o n must have the highest p r i o r i t y . - 6 -We w i l l return to discuss the problem of m u l t i p l i c i t i e s further a f t e r a more c a r e f u l exposition of the algebraic and geometric in t e r p r e t a t i o n s of the p r i n c i p l e of the conservation of number. Given an enumerative problem, l e t us assume that i t can be described by n homogeneous equations i n n+1 homogeneous unknowns. T h e o r e t i c a l l y we can eliminate variables one by one u n t i l we obtain a s i n g l e homogeneous equation i n two, homogeneous unknowns... The roots of t h i s equation correspond to the solutions of the o r i g i n a l system, thus the number of sol u t i o n s , counting m u l t i p l i c i t i e s , of the enumerative problem i s equal to the degree of t h i s equation. I t can be shown that t h i s degree i s the product of the degrees of the n equations i n the o r i g i n a l system which are independent of the c o e f f i c i e n t s . Thus the. (weighted) number of solutions to the enumerative problem i s conserved under continuous v a r i a t i o n of the parameters. However there are two snags. F i r s t l y , t h i s argument ignores the p o s s i b i l i t y of extraneous roots which could e a s i l y appear during the elimination procedure, and secondly, comparatively few enumerative problems can be described i n such a simple way. So, at the turn of the century, Schubert's calculus came under f i r e once again and, once again, i t survived. This time Giambelli (1904) and Severi (1912) rescued the calculus i n t h e i r papers both c a l l e d Sul p r i n c i p i o d e l l a conservatione d e l numero. In these papers Giambelli formulated and Severi developed the ideas that put the Schubert calculus on a geometric footing. Geometrically, an enumerative problem concerns conditions of i n t e r -section or tangency on figures of a c e r t a i n type, and though we are only interested i n a f i n i t e number, of these, i t i s us e f u l to look at the t o t a l i t y of a l l these f i g u r e s , f o r th i s set can be i d e n t i f i e d with a v a r i e t y . We say that t h i s v a r i e t y parametrizes the figures i n question, and we c a l l i t - 7 -the parameter v a r i e t y . Conditions imposed on the figures turn out to be algebraic ( i . e . defined by polynomial equations) i n an enumerative problem, so the set of solutions to the problem forms an algebraic set. A condition which reduces the freedom of the figures by r parameters, i s c a l l e d an r - f o l d condition and y i e l d s a subset of the parameter v a r i e t y of codimension r . Independent conditions correspond to subsets i n general p o s i t i o n , sum of conditions corresponds to i n t e r s e c t i o n of subsets, product of conditions corresponds to union of subsets, and equality of conditions corresponds to what we now c a l l numerical equivalence. Severi, i n his previously mentioned a r t i c l e of 1912 and i n h i s a r t i c l e Sui fondamenti d e l l a geometria numerativa. e s u l l a t e o r i a d e l l e  carat.teristiche of 1916, described the problem geometrically and developed an algebraic i n t e r s e c t i o n theory, but t h i s only solved the problem for i n t e r -sections of hypersurfaces on the parameter v a r i e t y . Some ideas of Poincare and Kronecker were developed by Lefschetz (1924, 1926) into 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 using simplices, and van der Waerden recognized that t h i s theory was s u f f i c i e n t l y general to give the Schubert calculus a rigorous foundation, and did so i n 1930 with h i s paper Topologische Begrundung des  KalkUls der abzahlenden 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 requires the d i f f i c u l t f a c t that to each algebraic subset can be assigned a class i n the cohomology of the parameter v a r i e t y . Two algebraic subsets i n the same continuous family are, h e u r i s t i c a l l y speaking, homotopic and consequently are assigned the same cohomology c l a s s . The i n t e r s e c t i o n of two algebraic subsets i n general p o s i t i o n i s assigned the cup product of t h e i r corresponding cohomology classes and t h e i r union the sum. It has also been shown that i f a f i n i t e number of algebraic subsets i n general 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 points - 8 -then the degree of the product of the corresponding cohomology classes i s equal to the number of points i n the i n t e r s e c t i o n , and consequently t h i s number does not change i f the algebraic subsets, i . e . the parameters of the problem, are varied continuously. Though t h i s constitutes a rigorous j u s t i f i c a t i o n of the p r i n c i p l e of conservation of number, inasmuch as we i n t e r p r e t the Schubert calculus within the context of the calculus of algebraic cohomology classes, we s t i l l cannot consider H i l b e r t ' s f i f t e e n t h problem solved. For i n the statement of the problem H i l b e r t makes i t clear that a l l the numbers, obtained by the c l a s s i c a l geometers have to be v e r i f i e d "with ah exact determination of the l i m i t s of t h e i r v a l i d i t y " and i n such a way that "the m u l t i p l i c i t y of t h e i r solutions may be foreseen." And so we return to the problem of m u l t i p l i c i t i e s . This problem has been stated i n modern terms and, i n theory, has been solved a b s t r a c t l y . 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 defined 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 , on the parameter v a r i e t y , of the algebraic subsets defined by the problem's conditions, at the point representing that s o l u t i o n . This d e f i n i t i o n has a l l the desired properties to solve any m u l t i p l i c i t y problem but i t i s d i f f i c u l t to do t h i s e x p l i c i t l y . This however would not s a t i s f y H i l b e r t since he requires the e x p l i c i t n e s s and not j u s t a general method. What i s needed then i s a set of general p r i n c i p l e s that w i l l deal with any m u l t i p l i c i t y without recourse to. any ad hoc methods i n a p a r t i c u l a r case. C l a s s i c a l l y , i t seemed that such a p r i n c i p l e was t a c i t l y assumed, and t h i s was that i n the general case of an enumerative problem ( i . e . where the figures are i n general position) each figure s a t i s f y i n g the prescribed con-d i t i o n s of contact i s counted with m u l t i p l i c i t y one. This seems to make i n t u i t i v e sense, i n fact i t almost seems to be a tautology, but i t s proof, i n terms of the preceeding formulation of the notion of a m u l t i p l i c i t y , i s - 9 -by no means t r i v i a l . The p r i n c i p l e does f a i l , as might be expected, i n p o s i t i v e c h a r a c t e r i s t i c . Kleiman has an example i n h i s paper The trans- v e r s a l i t y of a general translate, but he points out that t h i s example a r i s e s i n an unnatural way. Thus the p o s s i b i l i t y remains that some revised form of the p r i n c i p l e may be v a l i d i n any c h a r a c t e r i s t i c . Kleiman has proved that the p r i n c i p a l i s v a l i d ( i n zero c h a r a c t e r i s t i c ) unconditionally for l i n e a r spaces, moreover for any figures where the general l i n e a r group acts t r a n s i t i v e l y on the parameter v a r i e t y . For quadries, cubics and other higher-order figures however the problem remains unsolved. We do not even have complete knowledge of the v a r i e t y parametrizing complete twisted cubic space curves, for we lack the structure of i t s cohomology. r i n g . The problem of assigning m u l t i p l i c i t i e s i s deep, 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 done before we may.consider i t solved. In view of the preceeding remarks our example of the l i n e s meeting four given l i n e s i n 3-space i s p a r t i c u l a r l y nice, the more so because i t i s also easy to v i s u a l i z e . We now exemplify the algebraic and geometric interpretations of the Schubert calculus i n t h i s way. Preserve the notation above and l e t IK be any f i x e d plane containing L_^ . 3 If L meets any l i n e a r space X, i n IP the i n t e r s e c t i o n i s e i t h e r a point or a l i n e , or equivalently dim (Ln X) =0 or 1 r e s p e c t i v e l y . By convention, -1 i s the dimension of the empty set, so, i n p a r t i c u l a r , i f L and L' are skew l i n e s then dim (Ln L') = -1. The condition that L meet L., therefore l becomes dim (L n > 0. Now L i s not constrained to l i e i n any p a r t i c u l a r plane containing K and so, a p r i o r i , dim ( L n I K ) > 0. This however i s not an independent condition on - 10 -3 3 the set of l i n e s i n 3P . The f a c t that L i s contained i n IP , though, i s 3 an independent condition. I t i s expressed as dim (L n IP ) > 1 and means 3 i m p l i c i t l y that L i s not constrained to l i e i n a proper subspace of IP . More generally we can consider k-planes i n TPn and t h e i r i n t e r s e c t i o n s with subspaces of ]P n. The var i e t y parametrizing k-dimensional subspaces of IP n i s c a l l e d the Grassmannian and i s denoted by G^( 3 P n ) . Any condition imposed c l a s s i c a l l y can be formulated i n the following way. There i s a s t r i c t l y - i n c r e a s i n g , nested sequence, A o C A l C ' " * C \ 5 1 , 1 1 of l i n e a r subspaces of JP n such that any. k-plane X s a t i s f y i n g the imposed condition also s a t i s f i e s (Sch) dim (XnA.) > i 0 < i < k and v i c e versa. So a geometric condition on a k-plane i n IP n gives r i s e to k+1 independent algebraic conditions. It could be shown that the i - t h condition 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 conditions are considered together. Instead we show l a t e r that (Sch) i s an r - f o l d condition where k k r = V ( n - k + i ) - dim.(A.) = (k+1) (n-k) - V [dim ( A . ) - i ] 1=0 1 1=0 1 and that (k+1)(n-k) i s the dimension of G ( J P n ) . (Sch) i s c a l l e d a Schubert condition, and the set of a l l k-planes s a t i s f y i n g this condition i s c a l l e d a Schubert v a r i e t y which we denote by n[AQ,A^,...,A^J. It i s a v a r i e t y because i t s a t i s f i e s extra l i n e a r equations i n addition to the quadratic ones defining the Grassmannian which i s embedded i n pr o j e c t i v e space of 'n+1 !k+l dimension - 1. The Schubert v a r i e t i e s are then i n t e r s e c t i o n s of the - 11 -N Grassmannian with c e r t a i n hyperplanes i n IP . So the set of l i n e s i n 3-space meeting i s represented by the 3 Schubert v a r i e t y ^ [ L ^ , IP ] and so the set.of l i n e s meeting a l l four given l i n e s i s represented by the v a r i e t y 4 3 i = l 3 Now the parameter v a r i e t y i n th i s case i s G^( IP ) which has only one defining quadratic polynomial and thus i s a quadric hypersurface i n . Consequently V i s defined by one quadratic, and four l i n e a r equations. The elimination i s obviously 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 quadratic polynomial i n two homogeneous unknowns and therefore the number of l i n e s , i n general, meeting four given l i n e s i n general p o s i t i o n i n 3-space i s equal to the degree of t h i s polynomial which i s 2 x 1 x 1 x 1 x 1 = 2. Our hi s t o r y so f a r has brought us up to 1930 and van der Waerden's foundation of the Schubert calculus. With Ehresmann i n 1934 andi;his paper Sur l a topologie de certains espaces homogenes the calculus was put onto an even firmer foundation. He showed that the 2 i homology group of the >s Grassmannian with c o e f f i c i e n t s i n TL i s generated f r e e l y by the classes of Schubert v a r i e t i e s whose complex dimension i s i , (the odd dimensional groups are a l l t r i v i a l ) . For t h i s reason a Schubert v a r i e t y i s also \ referred to as a Schubert cycle. This i s the f i r s t part of what i s c a l l e d the basis theorem and at t h i s point we int e r r u p t the h i s t o r y . In the f i r s t two chapters we describe i n d d e t a i l the parameter v a r i e t y G^( IPn) and i t s Schubert subvarieties. We prove (both parts of) the basis theorem and show how the second part, which i s r e a l l y Poincare d u a l i t y , along with two formulae due to Giambelli and P i e r i put the Schubert calculus on a rigorous foundation by affor d i n g a complete d e s c r i p t i o n of - 12 -* n H (G iW ),7Z) as a 2 Z-algebra. In the l a s t chapter we return to our h i s t o r y , o u t l i n e some of the work done since Ehresmann and discuss the l i m i t a t i o n s of th i s and other i n t e r -section theories that have been developed since then. The problem of m u l t i p l i c i t i e s also occurs i n the theory of s i n g u l a r i t i e s of mappings which we mention b r i e f l y , as well as a d e s c r i p t i o n of the singular locus of a Schubert v a r i e t y wherein we include some of our own observations 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. A great many mathematicians have contributed to i t s p a r t i a l s o l u t i o n and t h e i r c o l l e c t i v e e f f o r t s have given b i r t h to new branches of mathematics, many of which have already born f r u i t . But there remain those parts that, i n t h e i r elusiveness, i n v i t e the conception of even newer theories. - 13 -Chapter I. THE GRASSMANNIAN §1 The Naked Grassmannian In studying geometrical objects that are "curved" one technique i s to consider a l l the best " s t r a i g h t " approximations i . e . the tangent spaces, and the question a r i s e s as.to where to put a l l these. And so we are led to consider the set of a l l k-dimensional subspaces of the ambient space. This object, however, i s so i n t e r e s t i n g i n i t s own r i g h t that we give i t a general foundation. For t h i s purpose, l e t E^ be a vector space of dimension n, then the set of subspaces of E of 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 fi x e d basis f o r E any k n n J ordered basis f or X gives r i s e to a k* n matrix over the ground f i e l d W which has rank k. We c a l l t h i s matrix the S t i e f e l matrix of the chosen basis, and the set of a l l such S t i e f e l matrices f o r a l l X e G, (E ) we c a l l k n the S t i e f e l space of k-frames i n E^, and denote i t by St(k,n). There i s an action of the l i n e a r group GL(k, W) on St(k,n) by l e f t m u l t i p l i c a t i o n . Since any o r b i t of t h i s action i s exactly the set of S t i e f e l matrices representing a l l the ordered bases of a given subspace i n G^(E_), the Grassmannian appears as t h i s quotient. H e u r i s t i c a l l y speaking, every point on the Grassmannian looks l i k e any other point. S t r i c t l y speaking there i s an action of Aut(E^) on E^ that induces an action on the k-subspaces of E^ which i s t r a n s i t i v e . Relative to a f i x e d basis f o r E we have Aut(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 basis vectors and E" i s the span of the remaining (n-k). If two automorphisms both take E' to X e G. .(E ) then t h e i r r a t i o leaves E' i n v a r i a n t , thus k n • G^CE^) can be r e l a b e l e d as the space of l e f t cosets of the i s o t r o p y group of E' i . e . GL(n, ] F ) / I s o t ( E ' ) . These two p o i n t s of view are r e c o n c i l e d as f o l l o w s : a matr i x A e GL(n, IF) sends E' to the subspace of E^ spanned by the f i r s t k columns of A, so l e t us w r i t e A = ( A ^ l * ) and define the map v i a m: GL(n, TF) — St(k,n) t where A f c denotes the transpose of A^. This map i s c l e a r l y s u r j e c t i v e . Consider the diagram GL(n, IF) c m *» St(k,n) (1.1.1) G. (E ) k n where ¥ i s the 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 to the span of i t s rows (or e q u i v a l e n t l y the o r b i t of the GL(k, TF) a c t i o n ) . I f (A^|*) g GL(n, W) then <f> o m( (A^| *)) i s the subspace of E^ spanned by the columns of A^, whereas ( ( A^ J * ) ) i s the coset (A^|*)Isot(E') which c l e a r l y represents the same subspace, so we have that (1.1.1) i s commutative. We note i n passing that IsotXE') c o n s i s t s of a l l matrices of the fo N r l -k - 15 -where 1^ e GL(k, IF), ^n_^ e GL(.n-k, IF), so Isot(.E') Is usually w ritten GL(k,n-k, IF) . A l l that we have dealt with so f a r i s the Grassmannian i n the l i g h t of " j u s t l i n e a r algebra," and though we w i l l continue to get more mileage from t h i s , we are s p e c i f i c a l l y interested i n examining structures that are derived when has a geometrical.foundation. In d i f f e r e n t i a l geometry then, we use the ambient spaces H n and (En and the corresponding G ( IRn) and G, (<Cn) have much i n common as d i f f e r e n t i a b l e manifolds. However most k of the geometric structure that we examine i s purely algebraic and for this reason we avoid the hassles of non. a l g e b r a i c a l l y closed f i e l d s . Henceforth then E i w i l l be the a f f i n e space of dimension n over IF, which we assume to be a l g e b r a i c a l l y closed, and so i n the p a r t i c u l a r case where IF = (E we are dealing with, two topologies on E^ ., the Z a r i s k i topology and the usual topology. With either topology on E^ = IF n the p r o j e c t i o n from the n-f o l d product of IF n with i t s e l f onto the f i r s t k factors i s a continuous open N 2 mapping. GL(n, IF) i s t o p o l o g i c a l l y . IF - R, where R i s the zero set of the determinant function therefore closed: i n both topologies. On the other kn hand we have St(k,n) i s t o p o l o g i c a l l y IF -S, where S i s the set of matrices with rank s t r i c t l y l e s s than k, but such a matrix i s characterized by a l l i t s k x k submatrices having zero determinant, thus S i s also closed i n both topologies. Certainly S i s contained i n the image of R under the projection, and so we get a new map GL(k, IF) > St(k,n) which coincides exactly with the map m i n diagram (1.1.1). Since R and S are closed, m i s also open and continuous. By the preceding argument and the - 16 -commutativity of (1.1;1) we deduce that Y and <J> induce i d e n t i c a l quotient topologies on G, (E ). k n We conclude t h i s section with the observation that when k = 1 the Grassmannian i s exactly P(E ), the proje c t i v e space associated to E , n showing that ^ ( E ^ ) i s a genera l i z a t i o n of one of the most important concepts of geometry. §2 The Grassmannian as Variety A great deal of the structure present i n the Grassmannian i s appreciated by seeing how i t presents i t s e l f as an algebraic v a r i e t y . As we have seen, a s p e c i a l case of the Grassmannian i s pr o j e c t i v e space, so one would not be surprised to f i n d out that the Grassmannians are 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 algebraic hold on the points of G,(E ) the ex t e r i o r powers K. n k come to hand e a s i l y . I f X i s a k-subspace of E n > then A X represents a " l i n e through o r i g i n " i n the vector space A E . Now, a basis n l k k, ' k k for 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 canonical mapping. p : G, (E ) + !P(AkE ) k n n which i s i n j e c t i v e . We show that the image i s closed with respect to the Z a r i s k i topology on 3P-(A^ E ) ; and thus G, (E ) clothes i t s e l f as a p r o j e c t i v e v a r i e t y . To n k n J see t h i s choose a basis for E say {e,,...,e } then e. A . . . A e . : n 1 n i , l ' 1 k k k 1 < i , < . . . < i , < h i s a basis f o r A E , so the points of TP (A E ) are k k n n represented by the i r homogeneous coordinates (...,x. . ,,..) r e l a t i v e to i 1 . . . i k k n thi s basis, and an a f f i n e open cover of IP (A E ) i s given by the (, ) sets n K. 17 -U . of points with homogeneous coordinate x. i r . . . . - , . i k 1 1 , , , m 1 . ^0. By elementary "k topology i t s u f f i c e s to prove that p(G (E )) nU. . i s Z a r i s k i closed i n U. L l ' Without loss of generality assume i . = j , and set U „ = U. Let 3 X , Z , . . . , K E' be the span of e 1 >...,e , E" the span of \+i>•••>en a n d x t h e s P a n o f ^,...,x^, then each x_^  has a unique representation e^ + e\| where e^e E', x eVe E". So x X ; L A . . . A x k = + e^ACe^ + e ^ ) A . . . A ( . e ^ . + e^ ') = e ' A e ' A . . . A e ' + (terms that are zero, or else not i n Thus p(.X) eU i f f e ^ A . . . A e ^ = X e ^ A . - . A e ^ for some \ 4 0, but then e^,...,e£ i s a basis for E' showing that we could have chosen the e_^'s o r i g i n a l l y to give X a unique basis of the form e^ + w^ , e^+w^,..., e^ + w^  where w. eE", This i s of course tantamount to choosing the a f f i n e coordinates u. = x. . /x .We have J -|_»" " *' k ^ l ' * " " ' ^ k 1>2,...,K „ ,s X , A . . . A X , = e 1 A . . . A e / + Y e A . . . AW. A.. . Ae, (1.2.1) 1 k 1 -.k i<i<k 1 1 fc + V e, A . . . A W . A . . . A W . A . . . A e , + . . . + W, A . . . A W . •* , , , 1 x i k 1 k l<x<j<k J where the w/s are a l l i n the place denoted by t h e i r subscript, showing that p(G,(E )) nU i s parametrized by the w.'s. These w.'s i n turn w i l l determine a kx ( n - k ) matrix B r e l a t i v e to the basis e. e , i n f a c t k+1 n the S t i e f e l matrix of x^,....,x^ i s ( i k l ' B ) = B i s determined - 18 -completely from the second term i n (1.2,1) by (1.2.2) u « = ( . - l ) k _ : L a... 1 < i < k l , z , . . . , i , . . . , k , j i j k+1 < j < n, where the circumflex over a subscript means that the subscript i s taken out. This has the e f f e c t of showing that p(G,(E )) i s covered by open sets each canonically isomorphic to a f f i n e space of dimension k(n-k) and the re s t of the a f f i n e coordinates are c l e a r l y related by polynomials to those of (1.2.2). Though some might be content to stop here, we s h a l l press on to fin d these r e l a t i o n s e x p l i c i t l y . F i r s t l e t us note that the a l t e r n a t i n g k k-l i n e a r form defined on X by the equation i = l P. (X) = P. (x ...,x ) = u. J l » , , , , : , k 3 l ' ' , - ' J k 2 V ' , 2 k i s independent of the c o e f f i c i e n t s of a l l the e.'s, save e. ,...,e. , i n the expansion of the x_/s and so i t i s a c t u a l l y a function of the rows of the k x k submatrix of the S t i e f e l matrix formed by taking the J-^ * • • • » j j ^ 1 columns. As such P. . must be a non-zero, scalar multiple of the J l ' , , , , J k determinant function; but th i s s c a l a r i s c l e a r l y independent of the rows chosen since P., ., w i l l perform, exactly the same sequence of arithmetic J l " * - , : i k operations on columns j ' ,.. . j ' to obtain x., ,, as P. . does on 1 k 3 l ' " " " ' ~ ' k i t s columns to obtain x. . . Denoting t h i s k x k matrix by A. . , J 1 >-.-»J k 2 V ' - > 3 k : i t follows from the observation of P^ ^ ^(x^,...,x^) = 1 that - 19 -(1.2.3) u. . = det(A. . ) 2 1 ' 2 2' * * *' 2k ~'l'*'"'~'k Expanding by minors along the row gives us (1.2.4) u. . . = J ("D a + 1a . detCA?' 1 . ) 2l'22'"'2k 1=1 a J i 2 V " , 2 k ct i where A.' . denotes the (k-1) x (k-1) matrix obtained from A. J-^» • • • » J - ^ > • • • > 1^ by deleting the a^1 row and i 1 " * 1 column. The a*"*1 column of A = 1L i s a l l zeroes except for a 1 i n the row, so cl e a r l y we have, on replacing the i ^ column of A. . b y the column of A1 „ , , j j > . . . » J k X,Z,...,K (1.2.5) detiA®'1 . ) = d e t ( A a , i . - : V " 3 k ll"-"2!^, A' where the circumflex beneath the subscript means that i t replaces the one taken out. Furthermore replacing the a*"*1 column of A. „ , by the i ^ i j Zj • • • j K column of A. .we obtain, as i n equation (1.2.2) J l 5 - - . . J k (.1.2.6) 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. . - (-1) u.. . , u. A . = 0 3j> "**'3k i = l J->...>ct,J^j...»K- j ^ , . . . , J ^ , ^ , - . . . j j ^ or i n homogeneous coordinates - 20 (1.2.7) 4- " x ,x. . - Y (-1)06 X x „ . x. A . = 0 X , . . . , K. J i . ' ' " * ' ~ ' k i = l x,. .. j 0 t j j ^ j . . . j l £ J 2> • • ' » J >• » • ' J j ^ where i t i s understood that the x. . are a l t e r n a t i n g i n t h e i r i n d i c e s . Thus we have shown that for any Xe p(G, (E )) n U there i s a point i n IP (A E ) k n n whose homogeneous coordinates s a t i s f y (1.2.7) for 1 < a < k, 0 < J 1 < J 2 < • • • < J k < n. Conversely we show that any point (...,x. . , ...) e IP(A E ) x^ > • • • > ^ k s a t i s f y i n g x^ ^ t u a n d equations (1.2.7) i s indeed a point i n p(G (E )) n U. Without loss of generality we may assume that x , = 1. k n X , • • • , K. Define a kx n matrix ( a ^ j ) ( a s i - n equation (1.2.2)) a i i = X l x k i ' But i f 1 < i < k then a.. =6.., the Kronecker d e l t a , thus (a..) i s the i j i j ' xj S t i e f e l Matrix ( |B) of a k-dimensional subspace X of E such that p n „ u(^) = !• Now consider the matrix formed from by replacing one of i t s columns by a column from B, then the determinant of t h i s matrix, k - i i . e . p. <» , . (x) i s simply (-1) a.., but since the r e s t of the X,...,X,...,K,J XJ coordinates p. . (X) are generated by these according to (1.2.7) then we have P. . (X) = x. V " , 3 k J l ' * * " ' J k for a l l sequences j^,...,jk« We summarize the above discussion i n the following theorem: - 21 -Theorem (.1.2.8) The mapping p : G, (E ) >-IP.(.A E ) i s a closed embedding, giving K n XI G^(E n) the structure of a non-singular p r o j e c t i v e v a r i e t y of dimension k(n-k). The P. . (X) are c a l l e d the Plucker.coordinates of X and J l ' " * , J . k p i s c a l l e d the Plucker embedding. This theorem allows us to make precise the notion of the Grassmannian as a parameter v a r i e t y . When we say that G^(E ) parametrizes k-planes i n n-space we mean that there i s a one-to-one correspondence between the set of k-dimensional subspaces of E^ and the p r o j e c t i v e var i e t y p(G, (E ) ) . Henceforth there i s no need to d i s t i n g u i s h between k n G, (E ) and i t s image under the Plucker embedding and so.we i d e n t i f y the two. A point 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 with rank (n-k). Since this system equally well describes conditions on the homogeneous coordinates of ^"(E n) y i e l d i n g the p r o j e c t i v e l i n e a r subspace 1P(X), G (E ) may also be thought of as parametrizing the ic n (k-1)-dimensional 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)-space. We write W B G k - i ( p ( E f i ) ) B G k - i { F n ' 1 ( I ) ) -The simplest 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 introduction. G„(E.) has dimension 4 1 2 4 and so i t i s a quadric hypersurface i n 3P^( IF) having the s i n g l e defining equation X12 X34 X13 X24 + X14 X23 " °' - 22 -In the next section we view the Grassmannian as a complex manifold. If IF = IR, G, (E ) is. a r e a l manifold also, but we do not discuss t h i s f o r k n reasons, mentioned before. In e i t h e r case, however, the Grassmannian i s compact, being a closed subset of p r o j e c t i v e space which i s compact. §3 The Grassmannian as Manifold The r e s u l t s of Sections 1 and 2 can be applied immediately to study the structure of G (E ) as a complex manifold. In the course of proving K. XI Theorem (1.2.8) we e s t a b l i s h that G, (E ) i s covered by open sets k n J W. = G (E ) n U. . which are a l l canonically isomorphic as J^>"'-»Jk k n -'l'*""'^k a f f i n e spaces of dimension k(n-k). These isomorphisms, i n the case where ]F = (C, are also biholomorphic and so G, (E ) i s the complex manifold associated to the algebraic v a r i e t y of Section 2. We can see these charts 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 St(k,n), which, now being an open subset of (C with the usual topology, takes i t s r i g h t f u l place among the manifolds. As i n Section 2, for any S t i e f e l matrix A l e t A. . be the matrix of columns . j . j then J 2»• • •»J k k the set V. . = {AeSt(k,n); det(A. . W O } J l ' •' • , J k 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 manifold which i s evidently stable under the a c t i o n of GL(k,(C) and so the image cp CV. . ) i s an open J l , , - , , J k set i n G (E ) which i s , of course, the set W. . above, k n 3 1' * * ' '^k As we have remarked i n Section 1, the Grassmannian i s the same a l l over. The group GL(n,(C) acts t r a n s i t i v e l y by automorphisms which are l i n e a r , whence algebraic and holomorphic, and so G^(E^) earns the t i t l e of a homogeneous space. This f a c t can also be seen i n a s l i g h t l y d i f f e r e n t way when we think of (Cn as endowed with i t s usual hermitian inner product. The t r a n s i t i v e a c t i o n i s now given by the unitary group U(n), and the isotropy of E' i s denoted U(k,n^k). Consider f i r s t the continuous map t : GL(n,(C) > GL(n,C) v i a A > AA A where A denotes the conjugate transpose of A. U(n) i s the inverse image under t of the closed set consisting of the i d e n t i t y t^, and so i s closed i t s e l f . From the equation AA = H for A = (z..) eU(n) we have, n i j n z . . = z . . z . . ^  V z . . z . . = 1 showing that U(n) i s both closed and bounded, i . e . compact. Since G, (E ) k n can be i d e n t i f i e d as U(n)/U(k,n-k) t h i s point of of view has the advantage of showing that the Grassmannian i s compact, without venturing into the algebraic category. We would, however, have ended up n a t u r a l l y at proje c t i v e space anyway, since i n t e r e s t i n g compact, complex manifolds can't l i v e i n an a f f i n e environment. It would be unwise to continue to separate the discussion into d i s t i n c t categories since part of the charm of the Grassmannian i s how the various structures flow into each other. Henceforth then we s h a l l assume t a c i t l y a l l the structure required by the context. - 24 -§4 The Universal Bundle over the Grassmannian Considering the Grassmannian as a complex manifold ( i . e . E^ = d n ) we can define a bundle over G, (E ) of rank k which has some useful properties. k n To each point X e G, (E ) we must associate a k dimensional vector space and k n the space X i t s e l f i s a natural choice. We must now demonstrate the existence of l o c a l t r i v i a l i z a t i o n s and show the compatability 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 efining the t r a n s i t i o n functions. To t h i s end r e c a l l the map 0 : St(k,n) -> G (E ) as defined i n section 1 K. n and the open cover {W^ .} as defined i n section 3 where I = ( i , . . . , i ). It i s easy to see that 3> i s holomorphic i n t h i s case. Denote by U the set K. c o n s i s t i n g of a l l pairs.(X,x) where Xe G. (E ) and x e X, so that k n U, c G, (E -) x C n . Define k - k n JJ : U, G. (E ) k k n v i a (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 wish k -1 to use (E as a canonical model of each of the f i b r e s TT (X) for each XeWj. Now, for a given X e W a S t i e f e l matrix A associated to X has the property that the k x k submatrix A^ formed by taking the i - j ^ ' ^2^' '^k^ columns i s non-singular, and we can without loss of generality assume A^ . = 3J^. A vector x e X i s a l i n e a r combination of the rows of A, and the coordinates of a vector Ve <E r e l a t i v e to the canonical b a s i s . The mapping : (X,x) -> (X>v) i s then the required one. On the overlap Wj. n Wj, X i s represented by a matrix A such that A^ . = l l ^ and such that A i s non-singular, where A i s the k x k submatrix formed by taking columns j ^ , . . . , J k - The matrix A, representing X and having A^ . = ll ^  i s - 25 -unique, likewise the matrix A- representing X and having A^ =. i . thus A = T A' where T = A e GL(_k, C) i s unique. The homeomorphism X X J F F " 1 : W n W x c k -> W n W x <ck J- J JL J J_ J i s given by (X,x) —• (X,T (x)) and the mapping t I J : w T n W J G L ( k ' c ) given by X —>• T i s evidently holomorphic. Thus U i s a holomorphic vector A. K bundle of rank k over G, (E ). k n We can define n global sections, S , of U, over G, (E ) as follows: l e t a k k n S T : WT WT x <Ek a, I I I be defined by S T(X) = (X,C ), where A i s as before and C i s the a, i- a a column of A. S i s c l e a r l y holomorphic and S. ,...,S. generate each a, I 1^,1 """k f i b r e over W^.. It remains to show that t h i s way of defining a s e c t i o n i s t r u l y global, i . e . that i t i s compatible with the t r a n s i t i o n functions on overlaps. I f XeW T also, then S T(X) = (X,C') where C' i s the a1"*1 column J a,J a a of A', and A' i s as before. But we see immediately that C = A C' a J a which i s a l l that i s required for the patching. The following theorem j u s t i f i e s the usage of the adjective " u n i v e r s a l " when r e f e r r i n g to U^. Theorem: Let M be a.complex manifold of dimension n. If K-> M i s a holomorphic vector bundle rank k, generated by n global sections r^,-. i .-,-r , then there i s a holomorphic map - 26 -<j> M -> G. (E ) T k n such that K i s the induced bundle A (U, ) and r . = A (S.). This r e s u l t i s T k 1 1 included f or the sake of completeness and i s re f e r r e d to only b r i e f l y , so we r e f r a i n from reproducing the proof here. The univ 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 denote the quotient bundle by Q ^ which has rank n-k and which i s c a l l e d the u niversal quotient bundle on G, (E ). Thus the sequence i£ XT 0 -»• U, G, (E ) x C n Q , -* 0 k k n n-k of bundles over G, (E ) i s exact, k n §5 The Dual Grassmannian If we consider f o r a moment a k-dimensional l i n e a r subspace X of 3Rn we see that there i s a unique (n-k)-dimensional subspace corresponding to X i . e . the orthogonal complement X^ r e l a t i v e to the usual orthonormal b a s i s . An isomorphism G, ( IRn) ^ G , ( 3R n) i s then obvious, but t h i s does k n-k not work for an a r b i t r a r y ground f i e l d and depends on a choice of bas i s . The idea that the set of k-planes i n n-space'should look l i k e the set of- (n-k)r-planes i n n-space can be formulated natu r a l l y as follows: l e t E = Horn ^ ( E , IF), where E i s a vector space of dimension n over an n JF n • n a r b i t r a r y ground f i e l d IF. Define, for XeG. (E ), • k n X° = {f e E : f (x) = 0, x e X} . n One checks that dim (X°) = n-k, so that we have a map It d : G ,(E ) -*• G (E ) k n n-k n which i s e a s i l y seen to be a set isomorphism. The inverse can be given by the map - 27 -G ,(£}-• G (I ) G, CE ) n-k n k n k n as defined by Y —• Y , where Y = { A e E : d>(f) = 0 f e Y} and where n i s o o n A the canonical isomorphism. G , (E ) i s called the dual Grassmannian. n-k n In the case where IF = <C, d i s a complex analytic isomorphism. Consider the universal bundle U , over G , (E ), the induced bundle n-k n-k n d (U , ) i s the dual of the universal quotient bundle on G, (E ). We have n-k k n the exact sequence of bundles on G , (E ), n—k n o+.u , -»• G . (E ) x en ^ q , 4 o n-k n-k n k _1 * ~ ~ and Q • = (d ) (U, ) where U, i s the dual bundle of U, . k k k k - 28 -Chapter II THE SCHUBERT VARIETIES §1 The D e f i n i t i o n We have up to now considered the Grassmannian as a completed form, but i t s true f a s c i n a t i o n l i e s i n s i d e . The alignment of the kr-dimensional sub-spaces with each other provides a means of c l a s s i f y i n g them even though each one was previously undistinguished by v i r t u e of homogeneity. To examine t h i s alignment we consider the f i l t r a t i o n on E^ determined by the chosen basis e, e , that i s 1 n (2.1.1) 0 = E c E , c . . . c E o 1 n where E . i s the span of e,-, ...,e.. We can think of E , as having the "best l 1 x k ° alignment" with t h i s f i l t r a t i o n and compare the other points X of G, ( E ) to k n E ^ by comparing the sizes of E ^ n E_^ and X n E_. . To t h i s end then we consider the sequence of integers j^dim (X n sequence of X and denote by i(.X) . In p a r t i c u l a r we have n \ which we c a l l the i n t e r s e c t i o n 1=0 i ( E ) =. (0,1,2,3,...,k,k,k,...,k) and t h i s i s our basic sequence. i(X) i s always a non decreasing sequence s t a r t i n g at zero and becoming eventually constant with value k. At each stage we allow one more dimension expansion, so i n t u i t i v e l y we should expect jumps i n the sequence of height at most one. This i s seen to be true by inspecting the pair of exact sequences i+1 x T 0 — • X n E . — • X n E . , , - 4 -— + JF i x+1 0 —> X n E — • x n E — X n E _ / X n E . —> 0 X X+X X+l X - 29 -where x^"*" i s the p r o j e c t i o n onto the ( i + l ) s t coordinate; that i s X n E . , , / X n E . has dimension 1 or 0 depending on whether or not X n E . , _ has l + l l 1+1 st any (i+1) coordinate. This argument shows that there are exactly k places where the dimension jumps. A general i n t e r s e c t i o n sequence then, looks l i k e C-0 J 0 J • • * j O y X j - X y a • • j X « 2 • 2 ^ • • • y 2 j • • • y X y l t —1 y a • • y lC~* X j lC j lC j a a a y k.) where the zeroth place i s always zero. Comparing i ( X ) to i ( E ) we. see that.the difference i s i n the place where the dimension jumps for the i ^ . time. For i ( E ^ ) the 1^ jump occurs at the th i place, but i n general there i s a lag of say, a^. In other words dim(X n E ) =. i but dim (X n E, , - -•) = i - 1 . These lags uniquely determine a .+i a +1—1 i l and, i n turn, are uniquely determined by i ( X ) . A property of i ( X ) i s that each integer 1 < i < k appears at l e a s t once, but the number of times i t does appear i s exactly ( a ^ + ^ + i+'l) - (a + i ) and thus a^+i~ a± ~ ®' Thus we have a b i j e c t i v e correspondence between the set of i n t e r s e c t i o n sequences and the set JJ, of sequences (a ,...,a ) such that J_ K. 0 < a^ < ... < a^ < n-k. We note here that the set mapping (a ,...,a ) — > (a^+1,a2+2,...,a^+k) gives a b i j e c t i o n between JJ and JK, the set of s t r i c t l y increasing sequences 1 < a| < a^ < ••• ^ a^ < n, so we see immediately that the 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 r \ n We can think of the sequence (a a ) as a measurement, i n some sense, of how awkwardly the k-plane X s i t s r e l a t i v e to the chosen f i l t r a t i o n on E. But we can also, use (a^,.,.ya^) as a bound on far we allow t h i s awkwardness to range as we vary the k-plane X , So we consider the set of k-planes whose th i n t e r s e c t i o n sequence has a .lag of at most a^ i n the p o s i t i o n of the i jump. Let us denote t h i s set fi('a^, . . ,, a^) . Equivalently, but more concisely - 30 -ft (a, , ,£1, ) - {Xg G, (E ) ; dim (XnE , .) > i , 1 <; i<; k . I k k n • a +i — • —' — 1 " This set can be described by relations among the Plucker coordinates x. . b y the following: X l ' • • • ' 1 k Proposition (2.1.2) fi(a^,...,ak) i s the subvariety of ^ ^(E^) corresponding to the linear polynomials x. . where j , . .., j i s any sequence such that j > a + X 1' " " k . A for some 1 <_ A <^  k. Proof: Let Xefl(a^,...,a^) and l e t j > a^ >-'X for some 1 <_ \ <_ k. Since dim(Xn E a +_^) >_ i , 1 <_ i <_ k we may. choose a basis, x^, . . . such that i 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 X11 X12 ••' X l , a i + 1 x, „x 00 X21 X22 x 2 , a + l ••• x2£ 2+2 00 XA1 XX2 ••• XA,a£fl *X^+2 +\ 00 kl^k2 '•• X k , a ; L + l Xk'a 2+2 "k.a^+X "k^+k 0 0 00 ... 0 consequently P. .. (X) i s the determinant of the matrix 31 -1 o x . . . X'\*-l X A + I , j 1 * * ' XA+1^A-I x . . ^•>31 X k ' J A - l x . k,3 x . . X . k , J k Using the Laplace expansion of the determinant we get d i r e c t l y that P. . (X) = 0, thus the l i n e a r polynomials a l l vanish on fi(a ,...,a ) Conversely, consider a point X of G, (E ) whose Plucker coordinates 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 coordinates on W -£•1 9***9 <£i_ 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 i > • • • »Ji ~r 'V"'Jk r = l that the point with Plucker coordinates x. .k - i whose S t i e f e l matrix i s (x. .) = ((-1) u„ has a basis x^,...,x ). Since u,, p t 0 then £. < a.+i. Now for any j > a . + i 1'"' * ' v i — we have k k r=l r=l r ^ i so by the maximality of Y., £ we get x. . =0. This shows that x s E r=l r 1 3 1 a i + : In the same way we get for i ' < i - 32 -l i thus dim ( X n E & +^) > i putting X i n f2(a . , . > a ) q.e.d. i Note that fi (a, > • • • »'a. ) e. G (E ) corresponds to the va r i e t y Q[A , ...,A. , ] C G , , (1P(E )) as defined i n the introduction where A. = P(E ,.) o Tc-1 - k-1 n I a.+x l and so we are completely 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 condition dim (Xn E, , .) > i 1 < i < k a.+i x i s c a l l e d a Schubert condition, which i s also consistent with 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 X K. symbol over G, (E ) . k n Some examples are i n order here. (2.1.3) n(0,0,...,0) In this case the k-planes are not allowed to roam at a l l , and so th i s v a r i e t y must consist of j u s t E^. To.see this properly note that, i n p a r t i c u l a r , dim (XnE^) > k for any Xgft(0,...,0). This must be an equality however, since both X and E have dimension k but 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 e fi(0, ... ,0, a, • • •, a) again i n p a r t i c u l a r dim (X n E^^) ^ k-d and so c.X, S i m i l a r l y since dim (X n E ^ ) > k we have x c E ^ + k . Conversely suppose E, , c g c E . For 1 ^ i ^ k-d E. C X thus J k-d a+k x dim (X nE.) > i , and for k-d+1 ^ i ^ k we have x dim (X nE , .) = k+cc+i - dim (X + E , .) > i ct+x a+i since X + E ^ ^ c Eo+k' S ° X s a t i s f i e s the requirements f o r l y i n g i n £2(0,. .., 0, a,.. ., a) , thus - 33 -n(P,...,0,a,...,a) = {X e G k C E n ) ; E ^ c X c E ^ } and by sending X to X/E k_^ we see that £2 (0, . . , ,0,a, , . • ,a). i s isomorphic to V W V d } = V V d ) • C2.1-.5). £2(n-k,n-k,. .. ,n-k) This v a r i e t y allows the la r g e s t possible lags, consequently the largest Schubert v a r i e t y . It i s i n f a c t the whole of G, (E ), since f o r any X e G, (E ) k n k n dim (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, therefore J n-k+x n-k+x~ dim (X E , .) > i and Xe£2(n-k,n-k,...,n-k). n—K+X §2. Example: The Schubert V a r i e t i e s i n G 2(E^) We now i l l u s t r a t e the previous section by attempting to v i s u a l i z e the geometry of the Schubert subvarieties of ^2^r? ' This Grassmannian, as we have mentioned before i s a quadric hypersurface i n TP'* with the si n g l e defining equation X 1 2 X 3 4 " X 1 3 X 2 4 + X 1 4 X 2 3 = °-The Schubert v a r i e t i e s are 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 that - 34 -52(0,1) = G 1CE 2) = JP 1, 9.(0,2) £ G 3(K 3) s JP 2, fiU.l) £ G 2 ( E 3 ) = G 1 ( E 3 ) = and from example (2.1.5) we have that £2(2,2) = G 2 ( E 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 to describe due to the fact that i t i s not smooth. I t i s the smallest example of a singular Schubert v a r i e t y , and the remainder of t h i s section i s devoted to i t s de s c r i p t i o n . In addition to the quadratic r e l a t i o n above, (2.1.2) t e l l s us that, on $2(1,2), we have the r e l a t i o n x^^-= 0, that i s J2(l,2) = V ( x 1 2 x 3 4 - x 1 3 x 2 4 + x 1 4 x 2 3 ) n V ( x 3 4 ) = V ( x 1 4 x 2 3 - x 1 3 x 2 4 ) n V ( x 3 4 ) . 5 3 Note that TP n V(x .) n V(x ) = JP where the homogeneous coordinates 9 1 » ^  are x 1 3 ' x 1 4 > x 2 3 a n d X24 a n d s o n ( x > 2 ) n v ( x 1 2 ) = V ^ X 1 4 X 2 3 — X 1 3 X 2 4 ^ n V ^ X 3 4 ^ 3 n V(x.„) i s a subvariety of JP . Since x,,x.„ - x,„x„, does not involve 12 14 23 13 24 x_. or.x,„ we can write t h i s v a r i e t y V'(x.,x„„ - x,„x„.). This i s simply the 34 12 14 23 13 24 1 1 ^ 3 image of JP x JP under the Segre embedding into our choice.of JP thus we have (2.2.1) £2(1,2) n V0x l 2) s IP 1 x JP 1 On the other hand £2(1,2) n {X e JP 5; x 1 2 f 0} = £2(1,2) i s a neighbourhood of (1,0,0,0,0,0) i n £2(1,2) and we can look at i t i n terms of the a f f i n e coordinates on W 2^. R e c a l l here the d e f i n i t i o n . o f the standard - 35 -open cover of G^ CE ) as defined i n Chapter I, sections 2 and 3, and set w = fi(.l,2)n W . For Xg W 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 A = f l 0 - x 2 3 - x 2 4 l ° 1 X 1 3 X 1 4 where x^^ = X^3 X24 ~ X]_4 X23 = "^ Since x^ 2 = det A^ 2 = 1 we can consider X13 ' X24' X14 a n c^ X23 a S a ^ i n e coordinates and as such i s a quadric 4 hypersurface i n C . The dimension of the Z a r i s k i tangent space at the orig i n i s 4, but everywhere else i t i s 3 showing that (1,0,0,0,0,0) i s an isolated singular point of £2(1,2). By a simi l a r analysis on each of the other open sets we can see that this i s , i n fact, the only singular point, thus (2.2.2) Sing £2(1,2) = £2(0,0) . For the remainder of':this discussion, then, we f i x the ambient space as 4 5 JP = IP n V(x^ 4) . We can now view £2(1,2) as the singular a f f i n e variety w^2> completed by i t points at i n f i n i t y , i . e . those lying 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 this set of points at i n f i n i t y i s the same as the Segre embedding of JPXx JP X into 3 4 JP = JP n V( x ^ ) . Since £2(1,2) i s covered by the sets , open i n £2(1,2), and since JPXx JPX n = 0 i t follows that (2.2.3) JP 1 x JP 1 = U w . . n V ( x l 9 ) . i=l,2 1 3 j=3,4 - 36 -This l a s t union can he broken down more u s e f u l l y as follows.: (2.2.4) ._V -w±j = W13 * [ W 1 4 X W 1 3 ] U [ W23 X W 1 3 J 1 1 [ w24 X ( w 1 4 U W 2 3 ) J x x, z j=3,4 where \ denotes the set difference. Relabel the sets on the r i g h t ;as D 1, D 2 ' D 3 a n d D4 from l e f t to r i g h t . To show that 4 i = l i s , i n f a c t a d i s j o i n t union i t s u f f i c e s to show that D x n D 4 = D 2 n D 3 = 0 but a point i n D 1 n must have = 0 = x^ and x^ 2 ^ 0, and a point i n D 2 n D 3 must have x^ £ 0 =f x^ and x^ = 0. Both of these are impossible since x ^ x ^ = x 1 3 x 2 4 ' Consider P"^  = C XL'{:OO} , where oo i s the point at i n f i n i t y then we have IP 1 x TP1 = (.C x C) XL '(W'x C) XL (C x {«>}) i i { (oo,*,)} . Again we r e l a b e l the sets on the r i g h t hand side as C, , -.C_, C_ and C, i n 1 2 3 4 the order that they appear. We wish to show that there e x i s t s such a decomposition s a t i s f y i n g (2.2.5) C . c D . for i = 1,2,3,4. x x • ? ? » F i r s t we choose l o c a l coordinates on each w. . for i = 1,2 and j =. 3,4. Let i j i j xi A be the S t i e f e l matrix of Xe w. . such that the 2x 2 submatrix A., of columns i and j i s the i d e n t i t y , then, by analogy with equation (1.2.2) we - 37 -have,: A 1 3 = '1 V l 0 A U = '1 V2 0 0 ' Q Z l 1 u l 0 Z2 U2 1 A 2 3 = f U3 1 0 o • A 2 4 = 1 0 0 ' Z3 0 1 U3 Z4 0 V4 1 where, setting v = ( X 1 2' X13' X14' X23' X24 )' x v = 13 1 v = A14 1 X23 1  X24 v = v = ( z i x> u r v r u i V ' ( z 2 , u 2, 1, u 2v 2, v 2 ) , ("z3> U 3 ' U 3 V 3 ' 1» V 3 } ' ( _ z 4 ' U4 V4' U4' V4' 1 } Since the aff i n e coordinates u,v and z on w „ are allowed to vary freely we see immediately that each w.. i s an affine space of dimension 3. In partic u l a r we have dim c (fl(l,2)) = 3. Restricting the l o c a l coordinates to w!. = w.. .n V(x ) reveals how (2.2.5 ) i j 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 W1 = w | 3 IL [w]_4 \wj_ 3 J JJ, [ w 2 3 \ w | 3 J j i {w\fi \ (w]A u w ; , ) J "24 x v"14 u"23' n D!, where D! = D. n V ( x ). . n x x x 12 x=l - 38 -Setting x = 0 amounts to k i l l i n g the z-coordinate on each w.., so we have Ll I J ' D! = x ?i; v1 0 0 0 0 1 u ; u v e c > = c - x c . In addition, s e t t i n g x = 0 amounts to k i l l i n g the u-coordinate on w', and X J 14 W23' s o w e a x s o n a v e D2 1 v 2 0 0 0 0 0 1 ; v 2 e c CX {co} and 0 1 0 0 0 0 1 v. ; v 3 £ C = W x C , F i n a l l y , s e t t i n g = x 2 3 = 0 amounts to k i l l i n g both the u and v coordinates on w^ ., and so the l a s t set i s D4 = 0 1 o o| 0 0 0 1 \ = {(»,»)} D! then i s the required set C. and we have established (2.2.5). - 39 *-Chapter I I I THE SCHUBERT CALCULUS § 1 I n t e r s e c t i o n Theory In t h i s s e c t i o n we summarize b r i e f l y the main ideas 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 theory developed by L e f s c h e t z , At the time of w r i t i n g , the book P r i n c i p l e s of 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 r e c e n t l y published. This book contains a complete, up-to-date v e r s i o n of t h i s theory, so any d e t a i l e d treatment here would be redundant. o o 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 manifold of dimension m. A s i n g u l a r p-chain C = on M, s a t i s f y i n g the property that each s i n g u l a r p-simplex, r, , i s the r e s t r i c t i o n to the standard p CO p-simplex A c JR of a C ! map from a neighbourhood of A to M i s c a l l e d P P a piecewise smooth p-chain on M. Since the boundary of a piecewise smooth D S chain i s piecewise smooth, we can de f i n e a chain complex C^ (M,Z) which i s a subcomplex of the s i n g u l a r chain complex. I t i s a f a c t from d i f f e r e n t i a l topology that the homologies of these two chain complexes are isomorphic. An (m-p)-cycle and an (m-q)-cycle are s a i d to i n t e r s e c t p r o p e r l y i f the i n t e r s e c t i o n has pure codimension p+q. They are s a i d to i n t e r s e c t t r a n s v e r s e l y at a p o i n t x i f the tangent space to the i n t e r s e c t i o n at x has codimension p+q a l s o . Let A and B be two piecewise smooth c y c l e s on M of complementary dimensions, i . e . dim A = p, dim B = m-p and l e t x e An B be a poi 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 , the tangent spaces T (A), T (B) to A X X and B at x are subspaces of 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 f a c t - .40 -T CM) = T (A) f T (B) , X X X Let {u 1 ?,,,,u } and {v.,,,,,v } be oriented bases for T (A) and T (B) 1 p 1 7 m-p x x respec t i v e l y then we define the i n t e r s e c t i o n index of A with B at x as follows det (u ,., . , ,u ,v ,,. , ,v ) i (A,B) = 1 . • • E_JL _ J t P _ _ det (u , , ,, ,u ,v ,,.,,v ) 1 p 1 m-p aat i s , + 1 according to whether or not {u.,...,u ,v.,.,,,v } i s an J- p 1 m«-D J- -^poriented basis for ( M ) . In the case where A and B i n t e r s e c t transversely everywhere we define <A,B> = I x (A,B) xeAnB x and c a l l i t the i n t e r s e c t i o n number of A with B . The word number i s j u s t i f i e d since, by hypothesis, dim (An B) = 0 so A n B i s a d i s c r e t e subset of M , which i s assumed to be compact. .Thus A n B i s f i n i t e . One shows that the i n t e r s e c t i o n number depends only on the homology classes of A and B , that i s i f A i s homologous to A ' then (3.1.1) < A , B > = < A \ B > , or, since {,) i s b i l i n e a r , < A , B ) ^ o for any A homologous to zero. For any two homology classes a e H ( M , 2Z) and g s H ( M , 2Z) i t i s possible ^ P m-p to f i n d two piecewise smooth cycles A and B representing a and 3 r e s p e c t i v e l y - 41 -such t h a t A and B meet t r a n s v e r s e l y on A n B , So we can d e f i n e a b i l i n e a r p a i r i n g on homology H (M, TL) x H (M, TL) + TL p m-p ' v i a (a , S ) + {a, B . ) = ( A , B ) , w h i c h i s c a l l e d the i n t e r s e c t i o n p a i r i n g o f a w i t h 3. The P o i n c a r e d u a l i t y theorem a s s e r t s t h a t t h i s p a i r i n g i s u n i m o d u l a r , i n o t h e r words t h a t Horn™ (H (M, TL) , TL) = { { a , ) ; a e H (M, TL)} , iL m-p P and a l s o t h a t i f , f o r a f i x e d et. e H P ( M J %•) , w e have ( a , g ) = 0 f o r a l l B s H (M, TL) t h e n a i s a t o r s i o n c l a s s . I n p a r t i c u l a r i f H (M, TL) i s f r e e m-p r p t h e n i s no n - d e g e n e r a t e . 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 t h e r e a r e isomorphisms D : H p(M,Q) -y H m _ p(M,4)) s a t i s f y i n g (3.1.2) (a.&) = [3,D(cO] ^jhere [ , ] denotes t h e K r o n e c k e r p r o d u c t . We w i l l f i n d (3.1.2) u s e f u l l a t e r . 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 t h e t o r s i o n . I n v i e w of the n e x t s e c t i o n t h i s i s p u r e l y a t e c h n i c a l i t y , We a r e now i n a p o s i t i o n t o p r e s e n t t h e main p o i n t s o f t h i s s e c t i o n . To t h i s end l e t M now be a compact, complex m a n i f o l d of d i m e n s i o n n over <E, so m = 2n. L e t V be an a n a l y t i c s u b v a r i e t y o f M of complex d i m e n s i o n 6, and s e t p = 2 3. L e t V have i t s n a t u r a l o r i e n t a t i o n . We can a s s i g n a cohomology c l a s s - 42 -to V i n the following way. Let a e p^s> ^ a n d choose a representative cycle A that meets V transversely i n smooth points, Again one shows that the intersection number <V,A> = I i (V,A) xeAnV i s independent of the choice of representative A for a, but this 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, We get around this problem by using the following t r i c k ; suppose A and A 1 are both representatives of a, then, using the fact 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 possible to find an m-p+1 chain C on M which does not intersect the singular 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 referred to G r i f f i t h s ' and Harris' book. V then defines a linear functional H (M, ZZ) -+TL, which by m-p ' J Poincare' duality i s of the form } for some yeH^(M, ZZ.) . This class or i t s Poincare dual D(y) g H m ^(M, ZZ) i s called the fundamental class of V. One might ask i f a l l t h is i s necessary, for a simpler way of assigning a cohomology class to V would be to look for a submanifold V' homologous to V, then assign V the fundamental class of V'. However t h i s , i n general, i s not possible. I t i s proved i n the paper Nonsmoothing of algebraic cycles  on Grassmannian var i e t i e s by Hartshorne, Rees and Thomas that the Schubert variety £2 (.2,2,3) i n GQ(E^.) i s not "topologically smoothable," - 43 -Suppose W i s an a n a l y t i c subvariety of M having complex dimension n ^9 and i n t e r s e c t i n g V transversely at x, Since there i s a natural choice for the orientations of V, W and M i t follows e a s i l y that l (.V,W) = 1, x This f a c t i s c e n t r a l to algebraic geometry. I t r e f l e c t s a basic difference between r e a l and complex geometry, If V and W i n t e r s e c t transversely i n a f i n i t e set of points then the i n t e r s e c t i o n number i s the number of points i n the s e t - t h e o r e t i c i n t e r s e c t i o n , whereas i n the r e a l case a cycle may cross another cycle, change d i r e c t i o n and cross back producing a c a n c e l l a t i o n . So, i n the r e a l case, the i n t e r s e c t i o n number i s usually somewhat smaller than the c a r d i n a l i t y of the i n t e r s e c t i o n . Now suppose that V and W do not i n t e r s e c t transversely. E s s e n t i a l l y t h i s means that some tangent d i r e c t i o n s to V and W at a point x e V n W have coalesced. By perturbing W s l i g h t l y inside i t s homology class we can count the l o s t tangents. This i s done l o c a l l y by f i n d i n g an a n a l y t i c v a r i e t y of dimension m whichiis a u-sheeted branched cover of a neighbourhood of x whose f i b e r over e i s the i n t e r s e c t i o n V n W + e. Here we think of £ e C n 3 v,W. m x(V,W) = u i s c a l l e d 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 of V and W at x,. and does not depend on the choice of l o c a l coordinates, If V and W i n t e r s e c t i n a f i n i t e number of points then, by an e x p l i c i t l o c a l analysis at each of the points x^, i t i s possible to f i n d a representative W' of the homology class of W which meets V transversely i n m (V,W) points i n the chosen neighbourhood of i x., Thus we have I (v,w) = {v,v:»} I m (V,W) , x. eVfiw ± X We note that a transverse 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 having u n i t m u l t i p l i c i t y , I t i s a simple matter now to see that the i n t e r s e c t i o n p a i r i n g i s Poincare dual to cup product i n cohomology, Consider the f o l l o w i n g diagram H m P(M,U)) x H P(M,q) ([M]n-) x i d Hp(M,<Q) x HP(M,«}) K i d . x D J H (M,OJ) x H (M.Q)-p m-p where D i s the Poincare 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 Kronecker product. Commutativity of the lower t r i a n g l e f o l l o w s from (3 .1.2) and the f a c t that f o r a n a l y t i c c y c l e s v e H (M, ZZ) , weH (M, ZZ) J P m - P 1 - X ( V , W ) = l x(w,v) = +1 consequently (v,w) = (w,v). [M] i s the fundamental c l a s s of M and n denotes the cap product so [M] n - i an a l t e r n a t i v e d e s c r i p t i o n of the Poincare d u a l i t y isomorphism. Since cup and cap products are a d j o i n t w i t h respect to the Kronecker product , commutativity of the upper t r i a n g l e f o l l o w s where C denotes the cup product. - 45 -I n case ex,(3 are two homology c l a s s e s not of complementary dimensipn we can s t i l l d e f i ne an i n t e r s e c t i o n product. Suppose a e H (M, 2Z) and • m-p g e (M, ZZ) , there 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 t r a n s v e r s e l y almost everywhere, We choose the o r i e n t a t i o n f o r C = A n B so that i f { v . . , . . . , V } 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, 1 m-p-q x and i f { U l " " ' V V l " " ' V p - q } a n d { v l " " ' V m - p - q ' W l ' " " W p } are bases f o r T (A) and T (B) r e s p e c t i v e l y then { U 1 " ' ' ' U q ' V • • " V m - p - q ' W l ' ' ' * ' W p } i s an o r i e n t e d b a s i s f o r T (M). 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 and we denote i t A-B, One checks that t h i s product i s w e l l - d e f i n e d on homology by f i n d i n g a chain 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 that 3D = A and then showing that 3(D * B) = A • B holds when O r i e n t a t i o n s are taken i n t o account. One a l s o checks that the product i s a s s o c i a t i v e , This product and indeed a l l c o f the preceeding 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 theory to the s u b v a r i e t i e s of the Grassmannian would be greatly^enhanced by an e x p l i c i t b a s i s f o r the homology. This was f i r s t described by Ehresmann who showed that the Schubert v a r i e t i e s (or more p r e c i s e l y t h e i r i n t e r i o r s ) provide a c e l l decomposition of ^ ( 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 chapter one. There we saw that 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, that i s the sequence (a^, . . . ,a^_) of lags i n the i n t e r s e c t i o n sequence. Therefore G. (E ) i s decomposed as a rC XI d i s j o i n t union of the sets (a^,...,a^) where u(a^,...,a k) = {XeG^XE^) ; (a^,...,a n) i s the l a g sequence of i ( X ) } = {XeG. (E ) ; dim (XnE ) = i , dim (XnE . . T ) = i-1} . K n a.+l a . + i - l '' l l This set i s the complement i n ft(a^,....a^) of the union of a l l i t s proper Schubert s u b v a r i e t i e s . By choosing 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 ; P r o p o s i t i o n (3.2.1) a f c) = JUa^ . . . ,afc) \ [ I fi(a1 a^a^.-l,...,^) ieN * where N = {1 < i < k; a. > a. ,} and where we set 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 holds f o r 1 < i < k then ft(b. , . ,, ,b. ) c ft(a. , , ,, ,a, ) . But i f b. < a. then the 1 k — 1' k i i decomposable k-yector e n A e A . . . A e . , , \ A e, , . , N A . , . A e , . A , , , A e 1 2 i-(a.-b.) b.+i+l a.+i a.+k i i i I l - 47 -s a t i s f i e s dim ( X n E , . ) =? i but not dim ( X n E, , .) > i , I t f o l l o w s that a. T I b . t i l l fi(b1? , , , ? b k ) c . fi(a1?, , , ,a k)< = >b i < a±, 1 < i < k and that the containment i s proper e x a c t l y when at l e a s t one of these i n e q u a l i t i e s i s s t r i c t . Thus u(a ,...,a,) i s non v o i d , X K. Suppose X e oo(a , , ,, ,a ) then dim (X n E n) = i - 1 < i thus X K. cl , T l ^ X X £ tt(a^,, . , . a ^ a -1, . . .a^) = £ 2 X f o r a l l i , Conversely i f X k £ 2 1 , f o r A a l l i , then dim (X n E & < 1. However i t i s a property of i ( X ) that i t has jumps of at most one. Thus dim (X n E ,.-,)= i - 1 and X e £2(a.. , . . . ,a, ) i q.e.d. Let us consider f o r a moment the example of £2(1,2) c G^CE^). (3.1.2) t e l l s us that a)(i,2) = £2(1,2) \ ( £ 2(1,1) u n(o,2)). £2(1,2) i s defined as a subvariety of G2(E^) by s e t t i n g the Plucker coordinate x ^ = 0. £2(1,1) and £2(0,2) have the 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 0 = x„. = 0 r e s p e c t i v e l y by (2.1.2). Since x_,x„„ = x-_x„, i t i s not 23 24 J 14 23 13 24 p o s s i b l e to v i o l a t e the co n d i t i o n s of membership i n £2(1,1) and £2(0,2) simultaneously by having x ^ ^ 0 4- and x^^ = 0. Thus X eo)(l,2) i f and only i f x^^ 4 0. We have then 03(1,2) = £2(1,2) n W 2 4 which g e n e r a l i z e s to - 48 -P r o p o s i t i o n (3,2,2) U ( . a 1 ? , , . ,a k) = B C a l f . . , ^  n W + k 1 ' ' * k Proof A poi 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 according to (2.1.2). Some of these are the d e f i n i n g r e l a t i o n s f o r £2(a^,, . , ,a^) , but the set of e x t r a El'ucker coordinates that vanish on S21 i s e a s i l y seen to be R. = {x. . ; j . = a.+i and j , < a, + A f o r X f i} . i J-L* -. . »J k i i > X Since x e | R. i t f o l l o w s that a k-plane X i n W ,.. a.,+1, . . . ,a, +k . x a.,+1, ... ,a,+k 1 k i e n 1 k cannot be i n any £21, thus 0 ( a l V 0 W a 1+l f . . . f a k+k £ u C ar--- ak )-I f X e u> (a, , . ..,a, ) then dim ( X n E , .) = i and dim ( X n E ,..,)= i - 1, 1 k a.+i a.+x-l I I so we can choose a b a s i s f o r X such that the i ' * 1 vector has a.+i1"'1 l coordinate non-zero. Therefore a S t i e f e l matrix A, f o r X d i f f e r s from that i n the proof of (2.1.2) only i n that each x. . i s guaranteed non-zero. X * Si . 'X 1 The k* k submatrix A . 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 zero 3 ^ T I j • • • , a ^ t K entry on the di a g o n a l , thus P a 1 + l , . . . , a k + k ( X ) = d e t \ + l * ° a n d X e W a 1 + l , . . . , a k + k ' ^ e ' d " We have already seen i n chapter 2, s e c t i o n 2 that £2(1,2) nW.. i s an a f f i n e space of dimension 3 over the complex .numbers, In p a r t i c u l a r there i s a homeomorphism - 49 -(0(1,2) £ JR6 e x h i b i t i n g to(1,2) as a 6 - c e l l , This a l s o g e n e r a l i z e s to P r o p o s i t i o n (3.2.3) 2(a +,.,+ak) u(a^,.,.,a k) i s homeomorphic to 3R Proof The S t i e f e l matrix A of X mentioned i n the proof of (3.2.2) has a. k p r i o r i £ a.+i e n t r i e s that are u n s p e c i f i e d but A i s not a unique r e p r e s e n t a t i v e of X. I f we choose A in s t e a d so that A ,, = I a^+1,...,ak+k k as u s u a l , or e q u i v a l e n t l y choose ^ ^a^+1,...,a^+k ^ then we s p e c i f y an e x t r a -^(k+l) e n t r i e s corresponding to those on and below the diagonal i n A M . There remain e x a c t l y a,+..,+a, e n t r i e s i n a^+1,...,ak+k 1 k A' that are f r e e to vary 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 that ti(a^, . . . ,a^_) i s i r r e d u c i b l e since i t i s the cl o s u r e of o i ( a ^ , . . . ,a^) which i s connected. So the Grassmannian G, (E ) i s the d i s j o i n t union of a f i n i t e number of k n J even dimensional c e l l s , and by (3.2.1) the cl o s u r e of any c e l l w (a. ,.. . ,a,) i s contained i n the 2(a,+. . .+a, )^-skeleton. Thus G, (E ) i s a f i n i t e CW 1 k k n complex, CW The CW chain group (G^(E^), ZZ) i s the fr e e a b e l i a n group on a l l the c e l l s w(a^,.,,,a^) such that a^+ , + = r . Since a l l the c e l l s are even dimensional every chain i s a c y c l e and no chain i s a boundary, so the homology groups are n a t u r a l l y isomorphic to the chain groups, A homology c l a s s depends only on the sequence of i n t e g e r s ( a ^ t l t f a ^ ) si n c e any two f u l l f l a g s E, E' of l i n e a r subspaces of ( i . e . any two bases f o r E ) are connected by an i n v e r t i b l e l i n e a r transformation from n •> ~ E^ to i t s e l f which induces 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 ransformation from JP(AE ) to i t s e l f c a r r y i n g o)(.a , , ,, ,a ) , defined r e l a t i v e to E, n x ic b i j e c t i v e l y onto u ' ( a ^ j i ' t > ^ ) » defined r e l a t i v e to E', We have proved The Basis Theorem, Part I The i n t e g r a l homology of G (E ) i s f r e e l y generated i n dimension 2r by JC n the Schubert symbols (a ,.,.,a ) where a +...+a = r . The odd-dimensional X K. X K. groups are a l l zero. So, f o r example, the homology groups of G^CE^) that are n o n - t r i v i a l have one generator (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 generators (0,2) and (1,1) i n dimension 4. Fi n d i n g the B e l t i numbers of G, (E ) i s a com b i n a t o r i a l problem. The k. n th 2r B e l t i number, g i s the number of Schubert c y c l e s (a ,...,a ) such that a^+a2+...+a^ = r . I f r < k then t h i s i s the number b(r ) of p a r t i t i o n s of r . When r > k the s i t u a t i o n i s not q u i t e so simple, but we can s i m p l i f y as f o l l o w s , l e t N = dim ( G j c ( E n ) ) = k(n-k) , we show that g 2 r = $2(N-r) " Let S, be the set of Schubert symbols defined over G, (E ) then the f u n c t i o n k,n k n f : S -y S defined by sending the symbol (a 1,...,a ) to the symbol iC f TL IC y n X iC (n-k-a^,...,n-k-a^) i s a b i j e c t i o n which r e s t r i c t s to a b i j e c t i o n between the set of symbols w i t h sum r and the set of symbols with sum k(n^k) - ( a n + ,..+a,) = N - r . Thus we have shown that B „ = B „ / X T x . Now " 1 k 2r 2(N-r) l e t N T = the gr e a t e s t Integer i n then f o r k < r < N ' B 2 r = t n e number of p a r t i t i o n s of r i n t o at most k p l a c e s , F i n a l l y , i f N' < r < N then N-r <_ N' and B 2 r = ^2(N-r) S^Yes u s those. - 51 -Since G, (E ) i s a compact, complex manifold the statement 3 n = 3„ / l T s k n • ' 2r 2(N-r) f o l l o w s more g e n e r a l l y from Poincare d u a l i t y . In f a c t we s h a l l see l a t e r that i f [ J i s theccohomology c l a s s of the Schubert symbol and [ ] denotes the dual c l a s s then [ ] * - I f ( )] §3. The Ring S t r u c t u r e i n Homology In the case where the f i g u r e s are 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 solved using the r e s u l t s of the previous two s e c t i o n s . F i r s t the s o l u t i o n set i s described as a zero-dimensional sub-v a r i e t y of the Grassmannian, that i s , w r i t t e n as an i n t e r s e c t i o n of a f i n i t e number of 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 ) . Since the d e s c r i p t i o n of the s o l u t i o n set u s u a l l y i n v o l v e s some degeneration i n t o s p e c i a l p o s i t i o n these points are counted w i t h m u l t i p l i c i t i e s . We have seen that the i n t e r s e c t i o n number then, counts the number of s o l u t i o n s i n the general 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 to f i n d the product of two Schubert c y c l e s as a l i n e a r combination of other Schubert c y c l e s . The f i r s t step i s to f i n d the i n t e r s e c t i o n numbers of p a i r s of cycle 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 that l k l k k n k k I b. = k ( n - k ) - I a. . 1=1 1 1=1 1 These v a r i e t i e s do not, i n general, 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 2(E 4) , We have codim £2(1,2) = 1, codim £2(0,1) = 3, I t f o l l o w s d i r e c t l y from the Schubert c o n d i t i o n s that - 52 -£2(1,2) n £2(0,1) = £2(0,1) and so codim (£2 (1, 2) n ft (0,1)) < codim ( £ 2(1,2)) + codlm ( £ 2(0,1)), I t w i l l be necessary then to 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^) f o r the c l a s s of ft (b^, .,, jb^) so-that the i n t e r s e c t i o n £2(.a^, , , , ,-a^ ) n ft ' (b^, , . , ,b^) i s t r a nsverse. We r e c a l l the f l a g E = (E. c E c . . . c E ) 0 1 n defined 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 de f i n e a f l a g 3E' = ( E ' c E ' c . . . c E ' ) 0 1 n where E'. i s the l i n e a r span of {e ,e }, so that the s u b s c r i p t x n-x+1 n s t i l l denotes the dimension. Now d e f i n e fl' (b. , .. . ,b. ) = {X e G, (E ) ; dim (X n E' .) > i , 1 < i < k} 1 k k n b.+x — — — x and consider fi(a^,...,a^) n ft' (b^,. .. ,b^) . I f X i s i n t h i s i n t e r s e c t i o n then i t s a t i s f i e s dxm ( X n E , .) > x and a.+x — x dim ( X n E ' .) > i , f o r 1 < i < k . D . t l — — — X A more convenient way of w r i t i n g the second c o n d i t i o n i s dim (XnE- ) , k - i + 1, k-x+1 Combining the two co n d i t i o n s i t f o l l o w s that (3.3.1) X n E „E» _ . + 1 ? M 0 ) , i s i s k x k-x+1 s i n c e T 53 -Y = I X n E a ] + [XHE- ] S X i k-i+1 thus: Y has dimension at most k« For C3.3.1) to h o l d , the d e f i n i t i o n s of E and E T f o r c e a.+i + b. .,.,+k-i+1 > n + 1 1 k-i+1 that i s a. +b. . > n - k , l k-i+1 ' arid by v i r t u e of the f a c t that ft(a_,,f.,a,) and ft (h ,. . . ,K ) are of 1 k 1' k complementary d intension :this becomes a.+b, .,. = n - k . l k-i+1 We have thus shown that s H a ^ , . . . ^ ) n fi'-(b , .. . ,bfc) i s empty unless b . = n - k - a. I have i k-i+1 a n d i n t^ l i s c a s e w e 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 . We E a . + i " E b k _ . + 1 + k - i + l = E a . + i " E n + l - ( a . + l ) ' which i s simply the l i n e a r span of e ... By (3.3.1) i t f o l l o w s that a .+i l j . i » e . 1 e , i ) i s a b a s i s f o r X and consequently dim ( X n E ,.) a,+1 a„+z a,+k ^ J a.+i i s e x a c t l y i . This puts X i n coCa.^ , , , . ,a k) so X i s a smooth p o i n t of ft(a^, , , , ,a k) , L e t t i n g "{v^, t , , ,v^ _} denote the l i n e a r span of y^, , , , ,v_., f i n d that - 54 dim ( X n E ' ,. = dim ( X p E ' , ) V 1 n' k'Vi +i + 1 = 1 and so X i s a l s o a smooth po i n t of 0,' (b , . , , ,b, ) . One shows that the X K. i n t e r s e c t i o n i s transverse. By denoting the c l a s s of n(a ,. ,.,a,) by the x tc Schubert symbol (a , ...,a,) we have shown that X K. {(a±,...,ak),(b15...,bk)} = 6* where I = (n-k-a k, n-k-a k_^, n-k-a^) and J = ( b ^ , . . . , b k ) . This i s the content of The Basis Theorem, Pa r t I I (a^,...,a k) = ( n - k - a k > n-k-a^^, n-k-a^) * where denotes the Poincare d u a l . This theorem allows us to f i n d the "coordinates" of an a r b i t r a r y 2 r - c y c l e (or i t s homology c l a s s ) r e l a t i v e to the b a s i s f o r the 2 r t ^ homology group. Let a be any c l a s s i n H (G (E ), ZZ), then by part one of the b a s i s theorem a " I 6 a 1 , , . . , a k ( a l ' " " a k ) where the sum runs over a l l Schubert symbols such that a^+,,P+ak = r . To f i n d 6'- we i n t e r s e c t both sides w i t h (n-k-a , , , , ,n-k^a 1) whose i n t e r s e c t i o n - 55 -number i s one w i t h (a^ ?,,,,a^) and zero w i t h any other c l a s s i n the sum. That i s 7a,(n-k-a ,,,«,n-k-a )\ = § The i n t e g e r s S were r e f e r r e d to by Schubert as the degrees of V i n a 1 , , . . , a k the case that a was the c l a s s of an i r r e d u c i b l e subvariety V of the Grassmannian. Suppose W i s another i r r e d u c i b l e subvariety of complementary dimension and l e t y, , be i t s degrees, that i s " • ^ > • • • » [ W J • ^ b i > - - - » b k ( b l ' " " , b k ) where [W] denotes the homology c l a s s of W and the sum ranges over sequences such that b,+. . .+b, = N - r . Then we have 1 k cL • • • c l 1''*"' k n-k-a^,...,n-k-a^ where again the sum ranges over a l l Schubert symbols (a^,...,a k) i n dimension r. In the case where k = l i . e . G, (E ) = P n ^ t h i s equation reduces to k n Bezout's theorem. The foundations of the Schubert c a l c u l u s are set down w i t h the b a s i s theorem together with the f o l l o w i n g two theorems which a l l o w us to compute the product of two a r b i t r a r y Schubert cyc l e s i n terms of a p a r t i c u l a r set c a l l e d the s p e c i a l Schubert c y c l e s . Let - 5 6 -a'(d) = n(n-k-1,,,,,n-k-l,n-k,,,,,n-k) where the number of ( n - k ^ l ) ' e that 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 of 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 k) • o d = £(1^, .. . ,bfc) where the sum ranges over a l l Schubert symbols ( b 1 , ... ,b ) s a t i s f y i n g 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.. b, ) = codim„ (a..,...,a.) + codim., a ( L l K. ( L l ; lc <L a k ' where coding (\^,...,\^) = 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 of (1,2) eH 6(G 2(E 4), 7L). By d e f i n i t i o n (1,2) = a2, and codim (1,2) = coding = 1. (1,1) and (0,2) are the only Scbubert symbols w i t h the d e s i r e d codimension and they both 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 formula. Hence (3.3.2) (1,2) (1,2) = (1,2) a i = (1,1) + (0,2). There i s a companion to P i e r i ' s formula which shows that H A ( G k ( E n ) , 2Z) can be generated as a r i n g by the cl a s s e s of s p e c i a l Schubert c y c l e s , w i t h the i n t e r s e c t i o n c y c l e as the product, I t i s - 57 ~ a a a 1 a Ca. 1' "'*' ak^ a 2 a,+k-2 a, k k where a. i s defined to be zero i f i < 0 or i > k. l Both of these theorems are 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 and H a r r i s . Their: treatment would be d i f f i c u l t to improve upon so we r e f r a i n from reproducing the proofs here. Let us apply our new found techniques to the simple enumerative problem of f i n d i n g the number of l i n e s simultaneously meeting four given l i n e s i n 3 general p o s i t i o n i n TP . We have already seen that the s o l u t i o n s are the poi n t s i n the subvariety 4 v = n ^ [ L , , J P 3 ] i = i 3 of G^(TP ) = G^iE^), where L^,L2,L^ and L^ are the four l i n e s . Now 3 ii[L^,TP ] i s the same v a r i e t y as £2(1,2), by the remark f o l l o w i n g (2.1.2), where L_^  = TP(E ^ ) and TP 5 = TP (E^) . Consequently i , the number of po 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 the i n t e r s e c t i o n number of the f o u r f o l d s e l f - i n t e r s e c t i o n of (1,2), that i s The l a s t expression f o l l o w s from (3,3.2) and can be computed as f o l l o w s ; i = ( 1 , 2 ) 4 « 4 C 1 , 2 ) 2 , C 1 , 2 ) 2 ) ' = -<(1,1) + C P,2),C1 ,1) + C0,2)}, - 5 8 -{(1,1) + (0,2),(1,1) + (0,2)} = <(1,1),(1,1)> + l{ (1,1), (0,2),} + { (0,2), (0,2)} and the second p a r t of the b a s i s theorem a p p l i e s k i l l i n g the second term, w h i l e the f i r s t and l a s t terms are both 1 s i n c e (1,1) and (0,2) are Poincare s e l f - d u a l . In agreement w i t h our previous s o l u t i o n s of t h i s problem we f i n d that there are two l i n e s i n general i n t e r s e c t i n g four given l i n e s i n general p o s i t i o n i n complex p r o j e c t i v e 3-space. Let us compute a higher dimensional example. We wish to f i n d the number of 2-planes i n p r o j e c t i v e 5-space that i n t e r s e c t 9 given 2-planes. The c o n d i t i o n that a 2-plane X meet a given 2-plane f[ i s dim (X n H 2) > 0 but since dim (X n I P 5 ) =2, then dim (X n TI-j) S 0 and dim (X n TJ.^ ) > 1 by a property of i n t e r s e c t i o n sequences, where ™ 5 n 2 - n 3 - n 4 - ' and dim n_ = i . Thus X l i e s i n the Schubert v a r i e t y fl[n2,n4, TP 5] = fiflP (E 3),TP (E 5),TP ( E 6 ) ] = £2(2,3,3) which i s the f i r s t s p e c i a l Schubert variety on G^CE^) = G 2(JP"'), The number we wish to compute i s equal to 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) , Applying P i e r i ' s formula,, the square of i s (1,3,3) + (2,2,3) = o2 + c2 where a 2 denotes the c y c l e (1,3,3), The t h i r d power of i s h ' CTi + a 2 • °i which by P i e r i ' s formula i s (0,3,3) + 2(1,2,3) + (2,2,2). R e c u r s i v e l y we f i n d 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 pa r t I I of the b a s i s theorem we can evaluate °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 general, that the number of k-planes i n n-space meeting h = (k+1)(n-k) general (n-k-1)-planes i s h!k!(k-1)!.•.3!2! , n!(n-1)!...(n-k)! and our two examples are s p e c i a l cases of t h i s , - 60 -Chapter IV MORE RECENT DEVELOPMENTS §1 The Hasse Diagram An object that contains a s u r p r i s i n g amount of in f o r m a t i o n about the 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 defined as the l a t t i c e associated to the set of Schubert v a r i e t i e s p a r t i a l l y ordered 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 of G. (E ) then the Schubert symbol (a 1,.,,,a.) defines an in t e g e r K. XI X K. point i n ]R . Define (a 1,...,a k) A (b 1,...,b k) = (min [ a ^ b ], min [a^.b ]) , (a^, .. . . a ^ . v (b 1 > . . . , b k ) = (max [a^b . ^ ] , min [ a ^ j b ^ ] ) . I t i s easy to check that 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 ,...,a1 ) i s a Schubert subvariety of G (E )} K,n x K X K K n i n t o a d i s t r i b u t i v e l a t t i c e . When d e a l i n g withi.the f i r s t model of H, , k,n A and v have geometric meaning i n that fi(a1}...,ak) A fl(b , ...,bk) = fi(a ,...,a k) n S2(b 1 > . . . ,bfc) id(a ,. .. ,a, ) v H(b , ..,,b,) = sm a l l e s t Schubert v a r i e t y c o n t a i n i n g J. K. X K. J2(.a1? , ,, ,a k) u fi(b1? , , «,bfc) , Before e x p l o r i n g the s t r u c t u r e and other models of H we draw the tC y n diagrams i n a few low dimensional cases, Many appealing patterns present themselves s t r a i g h t away. We deal with some that have geometric s i g n i f i c a n c e . The homology b a s i s can be picked out - 62 -(2 , £ , 3 ) 3.fc C 0,0,0) - 63 -e a s i l y , The generators f o r R^r (G^ (E^) , ZZ ) are the p o i n t s of ^ l y i n g on k the hyperplane x^+x^+ + x^ = r i n 3R , In our examples t h i s hyperplane 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 and the 4 3 5 generators are (1,1,1) and (0,0,2) or -R (G (E^) ,Z ) = ZZ 9 7L 9 ZZ w i t h generators (1,2,2), (1,1,3) and (0,2,3), e t c e t e r a . Thus the B e t t i numbers can a l s o be read e a s i l y i n the same way, so f o r example the even B e t t i numbers ( B ^ , ^ , • • • >®ig) o f G 3 ^ E 6 ^ a r e -*->l>2>3,3,3,3,2,l,l r e s p e c t i v e l y . The odd ones are, of course, a l l zero. In a l l the examples drawn, we see that the "top h a l f " and the "bottom h a l f " of the Hasse diagram have the same shape. This i s true i n general and k i s a consequence of Poincare d u a l i t y . The hyperplane II i n 1 V v - k< n- k> L i " 2 i = l 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. In t h i s case II n H k,n J C k n k,n i s the set of Schubert symbols that are Poincare s e l f - d u a l . A point above the hyperplane i s r e l a t e d to i t s Poincare dual below and t h i s p a i r of points defines a l i n e segment whose mid-point i s i n II. We do not, however, see any symmetry happening from l e f t to r i g h t i n general. In the case H and H and i n the case of H s and H , the diagrams are the r e f l e c t i o n s of each other i n a v e r t i c a l a x i s . This r e f l e c t i o n i s the Hasse diagram's i n t e r p r e t a t i o n of the c a n o n i c a l isomorphism G, (E ) •=  G , (E ) . We mention a t e c h n i c a l p e c u l i a r i t y here; even though k n n-k n J G, (E ) and G , (E ) are c a n o n i c a l l y isomorphic t h e i r corresponding Hasse k n n-k n 1 diagrams are not n e c e s s a r i l y the same, yet G , (E ) and G ,(£**), which are 0 J ' J n-k ' n n-k n ? not c a n o n i c a l l y isomorphic, have p r e c i s e l y the same Hasse diagram, Thus H depends only on the i n t e g e r s k and n, and H = H i f and only i f K. y Tl iC y Ti T1™*JC y Ti - 64 -k = n-k, witness 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 ' dual :but doesn't care /where the vector 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 to show how H and H f i t i n t o H^ ^ and how H^ ,. and H^ f i t i n t o H^ ^, More g e n e r a l l y the Hasse diagrams of G. (E ) and G . (E ) f i t i n t o H „ where m = max (k,n-k). This f i t i s a k n n-k n m,2m ' u s e f u l device to help understand the Hasse diagram's i n t e r p r e t a t i o n of d u a l i t y e x p l i c i t l y . The same argument used i n the l a t t e r p a r t of example (2.1.4) shows that fi(0, . .. ,0,a.., . .. ,a, ) of G. , , (E ,,) i s isomorphic to 1 k k+d n+d f!(a, a.) of G, (E ) where d i s the number of zeroes .in the f i r s t Schubert 1 k K. n symbol, This induces an i n j e c t i o n of sets H k , n ^ \ + d , n + d f ° r 3 1 1 1 ^ k ^ d> v i a (4.1.1) ( a l 5 . . . , a k ) (0,.. .,0,a ,...,a ) which preserves both bounds. I f we assume k < n-k then H , embeds i n a n-k,n d i f f e r e n t way i n t o a higher dimensional diagram. In f a c t H , —>• H . , f o r a l l 1 < k < n, 1 < d' n-k,n n-k,n+d ' v i a (a^, ,, ,, a^)•—>• ( a ^ , , , , , a^) which i s induced from co n s i d e r i n g an (n-k)^plane Y i n E cE ,, to be an * n n+d (n-k)-plane i n -E I f we now choose d to be the i n t e g e r such that ^k+d n+d = ^n-k n+d 1 , e * d = n*"2k we f i n d that the Hasse diagrams of the - 65 -Grassmannian and i t s dual are d i s t i n c t s u b l a t t i c e s of H , 0 , l N , n-k,2(n-k) The diagrams H\ ^ are p r e c i s e l y those which do d i s p l a y the "sideways" 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 : Hn-k,2(n-k) "* Hn-k,2(n-k) 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 of sets preserving both bounds. We wish i t to have the property that the image of H i s H , and that i t have order two. Such a map we c a l l a r e f l e c t i o n , K , n n — l c , n and i f CT eH „, , then we c a l l the image c the r e f l e c t i o n of o. To n-k,2(n-k) accomplish t h i s l e t r = n-k and N = 2(n-k) = 2r, choose a f l a g T£ = (EQ C E 2 c • • • c a n c* from i t define a f l a g 3D i n E ^ v i a D. = E° . i N - i where V° denotes the a n n i h i l a t o r of V. Let X be a poin t i n ai(a, ,...,a ) 1 r N o and consider i t s i n t e r s e c t i o n sequence { a ^ } ^ and l e t 3 ^ = dim (X n D j ) -We have a± = dim (X n E ) thus N - a ± = dim ([Xn E ^ 0 ) = dim (X° + E?) l = dim (X°) + dim (E?) - dim (X° n E?) = r + N - i - dim (X° n E°) therefore - 66 -I dim (X°n ) - a + ( r - i ) that i s S N - i ^ a i + ^ " ^ or, on r e p l a c i n g i by N-i (.4.1.2) 0. = a M . - ( r - i ) . Thus the i n t e r s e c t i o n sequence of X° w i t h respect to JD can be computed from the i n t e r s e c t i o n sequence of X w i t h respect to E . Denote the l a g sequence of {g.}1? _ by (b..,...,b ) and d e f i n e l 1=0 1 r A ( a i , . . ; , a r ) = (b ,...,b ) • The f a c t that the r e f l e c t i o n has order two i s c l e a r from the c o n s t r u c t i o n and thus the f a c t that i t i s a b i j e c t i o n of s e t s . That the bounds are preserved can be proved by t a k i n g a c l o s e look at the i n t e r s e c t i o n sequences and f o l l o w i n g i n d i c e s . The d e t a i l s provide l i t t l e i n the way of geometric i n t u i t i o n and are omitted here. G, (E ) embeds i n G (E ) as the Schubert v a r i e t y ft(0,...,0,n-k,...,n-k) where the number of zeroes i s n-2k. We compute i t s r e f l e c t i o n . Let 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) \ I, > V " • -y—• ' t t i-(n-2k) terms (n-k) terms k terms w i t h respect to IE and according to (4,1,2) - 67 -w i t h respect to B , Thus X°ew(k,k,,,,,k). Thus we have shown that the r e f l e c t i o n of (0,,.,,0,n-k,,,,,n-k), which represents G (E ), i s (k, , ,,,k) which represents G , (E" ), I t f o l l o w s that the r e f l e c t i o n map r e s t r i c t s to n-k n an isomorphism between H, and H , k,n n-k,n The r e f l e c t i o n map 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 (or, e q u i v a l e n t l y the homology considered as a r i n g w i t h the i n t e r s e c t i o n product). Our choice i s that of G r i f f i t h s , whereas others, notably Kleiman and Laksov, choose the s p e c i a l Schubert cycle s to be of the form (j,n-k,n-k,.,.,n-k) f o r 1 < j < n-k. So, where we have k s p e c i a l Schubert cycl e s on (G, (E ) the other choice K. n counts (n-k) s p e c i a l Schubert c y c l e s . Again by l o o k i n g at the i n t e r s e c t i o n sequences i t i s easy to see that these (n-k) a l t e r n a t e Schubert c y c l e s are simply the r e f l e c t i o n s of our choice of (n-k) s p e c i a l Schubert c y c l e s on G . fe ) . n-k n One would hope that G i a m b e l l i ' s formaula i s compatible w i t h the r e f l e c t i o n . Since c a l c u l a t i o n s i n higher dimension become very cumbersome very q u i c k l y we show the t r u t h of t h i s only i n the example of G^iE^). Our choice of s p e c i a l Schubert c y c l e s then i s = (1,2), = (1,1), whereas the other choice i s a-^  = (.1,2), = (0,2). Now (,2-ui2-A) = (y,A)* = aA CTA-1 y+l . y * aA ' au V l ' V l aA ' % s i n c e at l e a s t one of A - 1 and y + l i s outside the range s t i p u l a t e d by G i a m b e l l i ' s formula. Using the r e f l e c t i o n s i n s t e a d we have - 68 Note that codim a. x codim o. f o r i 1 1 ?2 and so by F i e r i ' s formula. o X • a a • A = a '•- a A ..y y A a y • a A = a In general P i e r i ' s formula i s preserved under r e f l e c t i o n s i n c e the only property 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, Returning to the Hasse diagram we c a l l a Schubert symbol a and immediate predecessor of 3 i n case a < 3 and there does not e x i s t y such that a < y < & where < i s the t o t a l o r dering on the Hasse diagram. Geometrically an immediate predecessor of 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 Since a)(a^,...,a^) i s smooth, the s i n g u l a r locus of ft(a^,...,ak) i s contained i n the union of i t s predecessors. We describe p r e c i s e l y which ones, but f i r s t l e t us look at the example of ft(l,2). In H . the symbol (1,2) i s the only one w i t h more than one immediate predecessor, moreover, ft(1,2) i s the only s i n g u l a r Schubert v a r i e t y in•-G (E^). This i s not a coincidence. In 1974, Svanes published 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 locus 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 proof of 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 only quote the r e s u l t , of ft (a , . . . ,a, ) . - 69 -Theorem (.4.1.3) Let k Q be zero and a^ > 1, I f k^ . i s defined i n d u c t i v e l y by a k . . < \ . • +1 " a k . ,+2 = = a k . < \ . + l l - l i - i i - i l I s-1 then Sing (£2 (a , ., ., a )) = M £2. where k = k 1 k j = l J £1 =fl(a ,...,a^ ,a - l , a 2-l, . . . ,a - l , a a 2,...,a) . 3-1 3-1 3-1 3 3 3 Note that the hypothesis a^ & 1 i s simply a convenience s i n c e To i l l u s t r a t e t h i s we consider an example l a r g e enough to see what i s happening, say Sing £ 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 that (2.2,2) i s a l s o a s p e c i a l case of (4.1.3) i . e . Sing £2(1,2) = £2(0,0) . A consequence of t h i s can be seen on the Hasse diagram, namely C o r o l l a r y (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 only i f the corresponding symbol i n the Hasse diagram has at l e a s t two immediate predecessors, - 70 -Proof: Suppose ( a , , , n ( a , l has at l e a s t two immediate predecessors then there e x i s t 1 < 1 < j < k such that a. T < a. = a . . < a., i - l i J - l J Consequently ft C a , a. . ,a .-1, . , . ,a . , - l , a . -l,a.,„,.,, ,a,) 1 5.1-1' i j - l j - l J+2 k i s contained i n the s i n g u l a r locus of ti(a^, , ,, ,a^) and i s non-empty. Conversely suppose (a..,..,,a,) has only one predecessor. Let a. be the X JC X f i r s t non-zero i n t e g e r i n (a^,...,a^) then the unique immediate predecessor i s C O ,...,0,a - l , a , . . . , a k ) . We c l a i m that a. = a. f o r a l l i > i , f o r i f not l e t a. be the f i r s t i n t e g e r i J i 5 i n the Schubert symbol (0, . . . ,0,a .,...,a, ) such that a. > a. then J k i J (.0, . . . ,0,a., . . .,a ,a -1, .. .,a k) would be a d i f f e r e n t immediate predecessor. So fi(a^,...,a^) i s of the form ft(0, . . . ,0,a,. . . ,a) which,bby example (2.1.4), i s the Grassmannian G..(E ,,) and i s th e r e f o r e smooth, q.e.d. In the course of the proof we have e s t a b l i s h e d C o r o l l a r y (4.1.5) A Schubert v a r i e t y i s smooth i f and only i f i t i s a Grassmannian. I t i s now easy to count the number of smooth Schubert v a r i e t i e s i n G,(E ). There are Q, (0, . . . ,0) and each ft(0, . .. ,0,a, . . . ,a) where a occupies the l a s t i places f o r a l l 1 < i < k and runs from 1 to (n-k). This makes k ( n - k ) + l smooth Schubert v a r i e t i e s , This a l s o gives us the curious f a c t that the number of s i n g u l a r Schubert v a r i e t i e s i s e x a c t l y the codimension of G k(E n) as a p r o j e c t i v e v a r i e t y v i a the Plucker embedding, At the time of w r i t i n g we see no i n t r i n s i c geometric reason f o r t h i s though we b e l i e v e there i s one, - 71 -The Plucker coordinates themselves, however, form another model of the Hasse diagram by l a b e l i n g the vertex (a , , r,,a.) as JX , and t h i s 1 K ^ l ' ' ' ' ? 3 ] j does have geometric content 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 point of w ( a 1 , , , , , a k ) i s a point of n (a , ,'f , ,afc) where x^ + 1 a +k ^ 0 ' 1 ' ' ' ' ' k I t i s p o s s i b l e to 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 , Poincare d u a l i t y commutes w i t h r e f l e c t i o n , i . e . the symbol A * (a^,.,.»ak) i s unambiguous. Suppose 1< X <... < A, ^ 2k and suppose I & 1 < y ^ < ... < < 2k i s the complementary sequence, by which we mean that ( y ^ , . . . , y k ) i s the set {l,2,...,n} \ {A^,....X^} arranged i n ascending order. Now suppose that (a ,...,a ) i s the ve r t e x of H- corresponding to X iC R ) Z K x and that ( b b , ) corresponds to x . We conjecture that A^. ••..,Ak 1 K- y ^ , . . . , y k A * (a 1,...,a k) = (b 1,...,b ) . I f I and J are complementary sequences then the quadratic r e l a t i o n k + X X = V (-1) P q x s X <" p=l 1 p ,^q k 1 q Ap k does not c o l l a p s e on W^. n W f o r 1 < q < k, thus the conjecture might help to f i n d independent quadratic r e l a t i o n s l o c a l l y , of which there should be ( k) - k(n-k) - 1 = the number of s i n g u l a r Schubert v a r i e t i e s . - 7 2 -§2 Concluding Remarks The i n t e r s e c t i o n theory used here r e l i e d h e a v i l y on the f a c t . t h a t we were working over the complex numbers. I t was developed by Lefschetz and ap p l i e d to the foundations of the Schubert c a l c u l u s by van der Waerden i n 1930. Ehresmann found the c e l l decomposition i n 1934 and developed some general r e s u l t s about c e l l complexes, that are now standard, to prove the ba s i s theorem. A l l of t h i s however was t o p o l o g i c a l . Hodge produced the f i r s t p urely a l g e b r a i c i n t e r s e c t i o n theory i n 1941 and 1942 w i t h the papers The base f o r a l g e b r a i c v a r i e t i e s of a given  dimension 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 formulae f o r a  grassmannian v a r i e t y . He proved the b a s i s theorem, and P i e r i ' s formula f o r an a r b i t r a r y s p e c i a l Schubert c y c l e o^, w h i l e van der Waerden had f i r s t shown the case where 1 = 1 . Hodge then used P i e r i ' s formula to prove G i a m b e l l i ' s formula. A great many i n t e r s e c t i o n t h e o r i e s were developed a f t e r Hodge's and i n the f o l l o w i n g we make no pretensions of completeness. Perhaps the most notable was the Chow r i n g , defined as f o l l o w s : l e t C (V) be the f r e e a b e l i a n 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 of V, a p r o j e c t i v e v a r i e t y over an a l g e b r a i c a l l y c l o s e d f i e l d . I f X and Y are 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 are s a i d to 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 every i r r e d u c i b l e component Z of X n Y. The ab 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 that was mentioned i n the i n t r o d u c t i o n was invented by.Se.rre i n 1965 i n a paper e n t i t l e d Algebfe l o c a l e - m u l t i p l i c i t e s , and i s defined i n terms of homological algebra. We denote the m u l t i p l i c i t y of a component Z of 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 . One now considers v a r i o u s equivalences on C (V) that guarantee choices of r e p r e s e n t a t i v e s f o r each p a i r of equivalence c l a s s e s , so that 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 equivalences and the strongest one.is c a l l e d l i n e a r or 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 uivalent i f they are both members of an a l g e b r a i c system of s u b v a r i e t i e s parametrized by TP \ The Moving Lemma For any two v a r i e t i e s X,Y-on V there i s a v a r i e t y Y', l i n e a r l y e q uivalent to Y, such that 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 are v a r i e t i e s such that X i s l i n e a r l y equivalent to Y then whenever X • Z and Y • Z are defined A X . Z i s l i n e a r l y equivalent to Y • Z. The quotient of C (X) by l i n e a r . equivalence i s the l a r g e s t r i n g f o r which the i n t e r s e c t i o n product i s defined everywhere. This r i n g i s c a l l e d the Chow r i n g . L i near equivalence can be weakened. Instead of having the continuous f a m i l y parametrized by we have i t parametrized by a q u a s i - p r o j e c t i v e v a r i e t y U. 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 equivalence. 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 homological equivalence means membership i n the same Weil cohomology c l a s s , where a Weil 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 zero f i e l d that 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 equivalence define'.d as f o l l o w s ; consider - 74 -C*(V) + Cn(.V) + 2Z n ' * where C (V) i s the subgroup of C (V) generated by subvarieties.of'codimension * n ( i . e . points)v, the f i r s t map i s the p r o j e c t i o n (C (V) i s a graded group), and the second map adds up the c o e f f i c i e n t s of the l i n e a r combinations: The second map i s c a l l e d the augmentation.' I f X • Y i s d e f i n e d , then i t s image under the composite map i s an i n t e g e r c a l l e d the i n t e r s e c t i o n number of X and Y. I t ignores components of X n Y of p o s i t i v e dimensions counting only p o i n t s w i t h t h e i r m u l t i p l i c i t i e s . I f <X,z) = <Y,z) <t'6v: every Z f o r which the product i s defined then X and Y are s a i d to be num e r i c a l l y e q u i v a l e n t . By analogy w i t h s e c t i o n 1 of chapter I I I one would expect homological equivalence =>numerical equivalence and indeed i t does, but the opposite 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 that there i s a c e r t a i n general c l a s s of v a r i e t i e s , which contains Grassmanmians and f l a g v a r i e t i e s , w i t h the property that numerical equivalence =>homological equivalence. He a c t u a l l y proved a more general r e s u l t but i t i s not needed here. Laksov, i n 1972, constructed an i n t e r s e c t i o n theory 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 Cycles on Grassmannian V a r i e t i e s . He proved, using t h i s theory, the b a s i s theorem and versions of both P i e r i ' s - 75 -and G i a m b e l l i ' s formulae. In co n t r a s t w i t h Hodge's-method, Leksov proved Gi a m b e l l i ' s formula f i r s t and used i t to prove P i e r i ' s formula. Using L e f s c h e t z ' i n t e r s e c t i o n theory we solved two examples i n enumerative geometry where the f i g u r e s were l i n e a r spaces. Kleiman has shown t h a t t h i s can always be done i n c h a r a c t e r i s t i c zero, and more g e n e r a l l y f o r any f i g u r e s where the general l i n e a r group acts t r a n s i t i v e l y on the parameter v a r i e t y . This i s not the case f o r conies; the general l i n e a r group i n t h i s case has four o r b i t s , namely the set of non-singular conies, the set of p a i r s of d i s t i n c t l i n e s , the set of double l i n e s w i t h d i s t i n c t f o c i and the set of double l i n e s w i t h double f o c i . In c e r t a i n cases however i t i s p o s s i b l e to solve an enumerative problem i f the f i g u r e s are not " s t r a i g h t . " For example, Kleiman and Laksov prove that the number ( o r i g i n a l l y found by Schubert) of 4 l i n e s common to two quadric hypersurfaces 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 that the number of l i n e s i n IP which simultaneously i n t e r s e c t four given curves C^, C^, C^, C^, i f f i n i t e , i s equal to 2 6 1 6 2 6 2 6 4 , where 6^ i s the degree of C^ and the number i s counted w i t h m u l t i p l i c i t i e s . The program given by H i l b e r t ' s f i f t e e n t h problem i s immense, there i s much c l a s s i c a l work s t i l l to v e r i f y . Some aspects of the problem have been solved repeatedly, but, i n t r u t h , we must s t i l l consider H i l b e r t ' s f i f t e e n t h problem unsolved. - 76 -BIBLIOGRAPHY 1. B o r e l , A., Lin e a r A l g e b r a i c Groups, W.A. Benjamin Inc., 1969. 2. Chern, S.S., Complex Manifolds Without P o t e n t i a l Theory, D. Van Nostrand Co. Inc., 1967. 3. Dold, A., Lectures on A l g e b r a i c Topology, Springer-Verlag, 1972. 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 Inc., 1969. 6. G i a m b e l l i , G.Z., "Sul p r i n c i p i o d e l l a conservatione d e l numero", Jahresb. deutsch. Math.-Ver. [13(1904), 545-556]. 7. Greenburg, M., Lectures on A l g e b r a i c Topology, W,A. Benjamin Inc., 1967. 8. G r i f f i t h s , P. and Adams, J . , Topics i n A l g e b r a i c and A n a l y t i c Geometry, P r i n c e t o n U n i v e r s i t y Press, 1974. 9. 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 of A l g e b r a i c Geometry, John Wiley and Sons, 1978. 10. Grothedieck, 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 " , Seminaire C. Chevalley E.N.S. (1958). 11. Hartshorne, R., Rees, E. and Thomas, E., "Non-smoothing of 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 " , B u l l e t i n of the American Mathematical Society [80, 5(1974), 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 of given dimension 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 of A l g e b r a i c Geometry, Cambridge U n i v e r s i t y Press, v o l . I I (1952), r e p r i n t e d 1968. - 77 -15. Kleiman, S.L., "Algebraic c y c l e s and the Weil c o n j e c t u r e s " , Dix  exposes sur l a cohomologie des schemes, North Holland (1968). 16. Kleiman, S.L., "The t r a n s v e r s a l i t y of a general t r a n s l a t e " , Compositio Math. [28, 3(1974), 287-297]. 17. Kleiman, S.L. "Problem 15. Rigorous Foundation of Schubert's Enumerative C a l c u l u s " , A.M.S. Proceedings of Symposia i n Pure Mathematics [28(1976), 445-482]. 18. Kleiman, S.L. and Laksov, D., "Schubert C a l c u l u s " , Am. Math. Monthly [79(1972), 1061-1082]. 19. Lakson, D., "Algebraic 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 Mathematical S o c i e t y , 1942. 21. M i l n o r , J . and Stas 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 Press, 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 pringer-Verlag, 1976. 23. Poncelet, 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. Porteous, I.R., "Simple s i n g u l a r i t i e s of 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 , Lecture notes i n math., Springer-Verlag [192(1971), 286-307] . 25. Schubert, H.C.H., K a l k u l der abzahlenden Geometrie, Teubner, L i e p z i g (1879) . 26. Serre, J.P., Algebre l o c a l e - m u l t i p l i c i t i e s " , Lecture notes i n math., 11, Springer-Verlag (1965). 27. S e v e r i , F., "Sul p r i n c i p i o d e l l a conservazione 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. S e v e r i , F., "Sui fondamenti d e l l a geometria 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 Veneto [75(1916), 1121-1162]. 29. Svanes, T., "Coherent cohomology on Schubert subschemes of f l a g schemes and a p p l i c a t i o n s " , Advances i n Math. [14(1974), 369-453]. 30. van der Waerden, B.L., "Topologische Begrundung des K a l k u l s der abzahlenden Geometrie", Math. Annalen [102(1930), 337-362]. 31. Zeuthen, H.G. and F i e r i , M., "Geometrie enumerative", Encyclopedie des sciences mathematiques, [ I I I , 2, 260-331] Teubner, L e i p z i g (1915). 

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