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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The Schubert calculus Higham, David Paul 1979
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Title | The Schubert calculus |
Creator |
Higham, David Paul |
Date Issued | 1979 |
Description | An enumerative problem asks the following type of question; how many figures (lines, planes, conies, cubics, etc.) meet transversely (or are tangent to) a certain number of other figures in general position? The last century saw the development of a calculus for solving this problem and a large number of examples were worked out by Schubert, after whom the calculus is named. The calculus, however, was not rigorously justified, most especially its main principle whose modern interpretation is that when conditions of an enumerative problem are varied continuously then the number of solutions in the general case is the same as the number of solutions in the special case counted with multiplicities. Schubert called it the principle of conservation of number. To date the principle has been validated in the case where the figures are linear spaces in complex projective space, but only isolated cases have been solved where the figures are curved. Hilbert considered the Schubert calculus of sufficient importance to request its justification in his fifteenth problem. We trace the first foundation of the calculus due primarily to Lefschetz, van der Waerden and Ehresmann. The introduction is historical, being a summary of Kleiman's expository article on Hilbert's fifteenth problem. We describe the Grassmannian and its Schubert subvarieties more formally and describe explicitly the homology of the Grassmannian which gives a foundation for the calculus in terms of algebraic cycles. Finally we compute two examples and briefly mention some more recent developments. |
Subject |
Calculus |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080156 |
URI | http://hdl.handle.net/2429/21372 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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