by
DAVID PAUL HIGHAM
B.Sc, Mount Allison University, 1973
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in
THE FACULTY OF GRADUATE STUDIES
Department of Mathematics
We accept this thesis as conforming
to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA
October 1979
David Paul Higham, 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study.
I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Department of Mathematics
The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5
Date October 1, 1979 ii
ABSTRACT
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. iii
TABLE OF CONTENTS
ABSTRACT . ii
TABLE OF CONTENTS iii
LIST OF FIGURES , , . . . iv
ACKNOWLEDGEMENT v
Introduction 1
Chapter I THE GRASSMANNIAN
§1 The Naked Grassmannian 13
§2 The Grassmannian as Variety 16
§3 The Grassmannian as Manifold 22
§4 The Universal Bundle over the Grassmannian 24
§5 The Dual Grassmannian 26
Chapter II THE SCHUBERT VARIETIES
§1 The Definition 28
§2 Example: The Schubert Varieties in G^E^) 33
Chapter III THE SCHUBERT CALCULUS
§1 Intersection Theory 39
§2 The Grassmannian as C.W. Complex 46
§3 The Ring Structure in Homology 51
Chapter IV MORE RECENT DEVELOPMENTS
§1 The Hasse Diagram 60
§2 Concluding Remarks 72
BIBLIOGRAPHY 76 iv
LIST OF FIGURES
Figure 1 ^2 3 61
Figure 2 ^ 61
Figure 3 ^2 h 61
Figure 4 H3 5 ^
Figure 5 R2 5 61
Figure 6 HQ a 62 V
ACKNOWLEDGEMENT
I am indebted mainly to Larry Roberts, my advisor, for his patience in seeing this work through to its com• pletion. I would also like to thank Jim Carrell for the original idea and for many of the useful references.
Gratitude is also due to Roy Douglas, Mark Goresky, Jim
Lewis, Ron Riddell and Bill Symes for their sympathetic and sometimes inspiring discussions.
Worthy of mention also are those who helped in a non-professional capacity by providing encouragement, moral support, tea and sympathy. Those uppermost in my mind are Roy Douglas, Ed Granirer, Fred Henry, Mike Margolick,
Ken Straiton and Scott Sudbeck.
Finally, a word of thanks to the "behind-the-scenes" people Mrs. MacDonald and Kathy Agnew for their many kindnesses and bureaucratic short-cuts, and Mrs. Janet
Clark for her intelligent typing of the manuscript. - 1 -
INTRODUCTION
During the last century work in geometry was highly intuitive. This was especially true of the so called enumerative geometry, which attempted to answer the question "How many figures in general satisfy a prescribed set of geometric conditions?" A simple example of this is to find the number of lines that meet four given lines in general position in 3-space.
Poncelet began work on questions of this nature while in a Russian military prison at Saratow in 1813. He published a paper entitled Traite des proprigtes projectives des figures in 1822 in which he introduced a notion called the principle of continuity. Roughly put, the principle states that the number of solutions to an enumerative problem does not change if the parameters are varied continuously. The principle was not properly justified, and Cauchy criticized it seriously before the paper was even published. In spite of Cauchy's influence, which created some prejudice, the principle obtained widespread popularity and the resulting controversy has not been completely resolved even to this very day.
Hermann Casar Hannibal Schubert was a prolific geometer and, having revived the principle, used it to calculate the solutions to an astounding number of enumerative problems. His fertile mind produced numbers that were often in the tens and hundreds of thousands or more, long before the advent of the modern electronic computer, though, ironically, after the development of the ill-fated "difference engine" of Charles Babbage.
In 1874 Schubert changed the name of the principle to the principle of special position in an attempt to avoid the prejudice. Schubert, however, was not satisfied that this name embodied the notion of continuous variation, and so the principle received its final baptism, two years later, as the principle of conservation of number. Despite the wealth of his - 2 - contributions to enumerative geometry though, Schubert realized that the principle still needed to be confirmed.
Returning now to the example mentioned above we will see how Schubert answered the question. Let the four lines be L^, L^, L^ and L^ and assume that they are in general position. Now move L^ so that it intersects L^ at P, and move L^ so that it intersects L^ at Q. The lines are now in
"special position" and it is easy to count the lines that pass through all these four lines. One line, L, is defined by P and Q, and since each pair of intersecting lines spans a plane, the line of intersection L', of the two planes is a second line passing through all four lines in special position. Suppose there is a third line L". To avoid notational clumsiness we will denote the specialized lines by L^ and L^ also. Now let R. be.the point of intersection of L" and L.. Since. L" is distinct 1 1 from L, L" does not pass through both P and Q. Thus if L" passes through
P, say, then as Q is the only point common to L^ and L^, R^ and R^ are distinct, i.e.
contains at least three points. First suppose that A = {P,R^,R^} (or equivalently {QJR^JR^}). The line defined by R^ and R^ (i.e. L" itself) lies in the plane spanned by L^ and L^, forcing P to lie in that plane also. But then the two planes must be coincident, in which case there would be an infinite number of lines passing through the four given lines.
Secondly suppose that the R^ are all distinct; then L" lies in both planes, and since L' and L" are distinct this again forces the two planes to be coincident. - 3 -
Schubert then brought into play the principle of conservation of number, which rested on a weak foundation, to conclude forthwith that the number of lines meeting all four given lines remains two when and are returned to general position, provided of course that the number is finite in the first place. Incidentally, another degenerate case to avoid is the possibility of all 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 it he computed number after number of solutions to enumerative problems. All 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 still 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 is 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, in principle, the
possibility of carrying out the processes of elimination, yet
for the proof of the theorems of enumerative geometry decidedly
more is requisite, namely, the actual carrying out of the process
of elimination in the case of equations of special form in such
a way that the degree of the final equations and the multiplicity
of their solutions may be foreseen.
Poncelet had, in 1822, claimed, that the principle could be verified
algebraically but didn't do so because he felt that the problem should be
viewed purely geometrically. Schubert felt the same way, though he stated
in his book that if the principle were interpreted algebraically it would
amount to saying that the number of roots of an equation doesn't change if
the coefficients are varied. Earlier, in 1866, de Jonquieres had tried to
establish this by applying the fundamental theorem of algebra. But since
a polynomial can have repeated roots we might expect to have multiplicities
to contend with sometimes., and indeed such is the case as, returning to our example, we now demonstrate.
Suppose after specializing the lines that the plane of L^, and is parallel but distinct from the plane of and L^, then rather than say
there is no line passing through all four lines, we allow the solution at infinity. To have the complete picture we also want to include imaginary points, and so the ambient space is complex projective 3-space.
We have ruled out the possibility of no solution but we cannot rule out the possibility of only one solution as the following will show.
Choose L^, and to be three skew lines 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 intersect nor are they parallel then H0 and are distinct. Now choose L, to be n..n:ll-i let P. = L. n L. for i = 1,2,3 and these points 4 2 3 i 4 1 ' are distinct since the three lines are skew. L, is a line that meets all 4
four.lines L^, so now assume that L is a different line passing through all
L_^. Since is 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 lies in the plane defined by them, but since L. and L. both lie in n~ this plane is exactly n„• Thus we have 2 4 2 2
(ji ^ Ln '£112^1"^ = {P-j} i.e. L meets at P^. Similarly L meets at
P^. But then L is defined by P^ and P^ and so must be itself, contra•
dicting the hypothesis that L and are distinct. Thus is the unique
line meeting all four given lines.
The principle of conservation of number can still be salvaged so long
as we count with multiplicity two. At first this may seem somewhat
contrived until we remember that this configuration of lines is actually
a degenerate case where, the two solutions of the general case have
coalesced into one of the four lines, namely L^. The principle starts to
become clarified when we state it like this: if the number of solutions
to an enumerative problem is finite then that number, counted with
multiplicities in the special case, is the same as the number of solutions
in the general case.
The problem of counting multiplicities is decidedly difficult, though
it was approached with great courage, and multiplicities were assigned with great alacrity by the adept classical geometers. This problem is
central to the rigorous foundation of Schubert's enumerative calculus, in
fact, in their article GEometrie Enumerative of 1915, Zeuthen and Pieri consider it of such fundamental significance that they state that obtaining its solution must have the highest priority. - 6 -
We will return to discuss the problem of multiplicities further after a more careful exposition of the algebraic and geometric interpretations of
the principle of the conservation of number.
Given an enumerative problem, let us assume that it can be described by n homogeneous equations in n+1 homogeneous unknowns. Theoretically we can eliminate variables one by one until we obtain a single homogeneous equation in two, homogeneous unknowns... The roots of this equation correspond to the solutions of the original system, thus the number of solutions, counting multiplicities, of the enumerative problem is equal to the degree of this equation. It can be shown that this degree is the product of the degrees of the n equations in the original system which are independent of the coefficients. Thus the. (weighted) number of solutions to the enumerative problem is conserved under continuous variation of the parameters.
However there are two snags. Firstly, this argument ignores the possibility of extraneous roots which could easily appear during the elimination procedure, and secondly, comparatively few enumerative problems can be described in such a simple way. So, at the turn of the century,
Schubert's calculus came under fire once again and, once again, it survived.
This time Giambelli (1904) and Severi (1912) rescued the calculus in their papers both called Sul principio della conservatione del 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 inter• section or tangency on figures of a certain type, and though we are only interested in a finite number, of these, it is useful to look at the totality of all these figures, for this set can be identified with a variety. We say that this variety parametrizes the figures in question, and we call it - 7 - the parameter variety. Conditions imposed on the figures turn out to be algebraic (i.e. defined by polynomial equations) in 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, is called an r-fold condition and yields a subset of the parameter variety of codimension r.
Independent conditions correspond to subsets in general position, sum of conditions corresponds to intersection of subsets, product of conditions corresponds to union of subsets, and equality of conditions corresponds to what we now call numerical equivalence.
Severi, in his previously mentioned article of 1912 and in his article
Sui fondamenti della geometria numerativa. e sulla teoria delle carat.teristiche of 1916, described the problem geometrically and developed an algebraic intersection theory, but this only solved the problem for inter• sections of hypersurfaces on the parameter variety. Some ideas of Poincare and Kronecker were developed by Lefschetz (1924, 1926) into a topological intersection theory using simplices, and van der Waerden recognized that this theory was sufficiently general to give the Schubert calculus a rigorous foundation, and did so in 1930 with his paper Topologische Begrundung des
KalkUls der abzahlenden Geometrie.
A topological intersection theory first requires the difficult fact that to each algebraic subset can be assigned a class in the cohomology of the parameter variety. Two algebraic subsets in the same continuous family are, heuristically speaking, homotopic and consequently are assigned the same cohomology class. The intersection of two algebraic subsets in general position is assigned the cup product of their corresponding cohomology classes and their union the sum. It has also been shown that if a finite number of algebraic subsets in general position intersect in a finite number of points - 8 - then the degree of the product of the corresponding cohomology classes is equal to the number of points in the intersection, and consequently this number does not change if the algebraic subsets, i.e. the parameters of the problem, are varied continuously.
Though this constitutes a rigorous justification of the principle of conservation of number, inasmuch as we interpret the Schubert calculus within the context of the calculus of algebraic cohomology classes, we still cannot consider Hilbert's fifteenth problem solved. For in the statement of the problem Hilbert makes it clear that all the numbers, obtained by the classical geometers have to be verified "with ah exact determination of the limits of their validity" and in such a way that "the multiplicity of their solutions may be foreseen."
And so we return to the problem of multiplicities. This problem has been stated in modern terms and, in theory, has been solved abstractly. The multiplicity of a solution is defined as the intersection multiplicity, on the parameter variety, of the algebraic subsets defined by the problem's conditions, at the point representing that solution. This definition has all the desired properties to solve any multiplicity problem but it is difficult to do this explicitly. This however would not satisfy Hilbert since he requires the explicitness and not just a general method.
What is needed then is a set of general principles that will deal with any multiplicity without recourse to. any ad hoc methods in a particular case.
Classically, it seemed that such a principle was tacitly assumed, and this was that in the general case of an enumerative problem (i.e. where the figures are in general position) each figure satisfying the prescribed con• ditions of contact is counted with multiplicity one. This seems to make intuitive sense, in fact it almost seems to be a tautology, but its proof, in terms of the preceeding formulation of the notion of a multiplicity, is - 9 -
by no means trivial. The principle does fail, as might be expected, in
positive characteristic. Kleiman has an example in his paper The trans-
versality of a general translate, but he points out that this example arises
in an unnatural way. Thus the possibility remains that some revised form of
the principle may be valid in any characteristic.
Kleiman has proved that the principal is valid (in zero characteristic)
unconditionally for linear spaces, moreover for any figures where the
general linear group acts transitively on the parameter variety. For
quadries, cubics and other higher-order figures however the problem remains
unsolved. We do not even have complete knowledge of the variety
parametrizing complete twisted cubic space curves, for we lack the structure
of its cohomology. ring. The problem of assigning multiplicities is deep,
as is Hilbert's fifteenth problem itself, and there is much to be done
before we may.consider it solved.
In view of the preceeding remarks our example of the lines meeting
four given lines in 3-space is particularly nice, the more so because it is
also easy to visualize. We now exemplify the algebraic and geometric
interpretations of the Schubert calculus in this way.
Preserve the notation above and let IK be any fixed plane containing L_^.
3
If L meets any linear space X, in IP the intersection is either a point or
a line, or equivalently dim (Ln X) =0 or 1 respectively. By convention,
-1 is the dimension of the empty set, so, in particular, if L and L' are
skew lines then dim (Ln L') = -1. The condition that L meet L., therefore l becomes dim (L n > 0.
Now L is not constrained to lie in any particular plane containing K and so, a priori, dim (Ln IK) > 0. This however is not an independent condition on - 10 - 3 3 the set of lines in 3P . The fact that L is contained in IP , though, is 3 an independent condition. It is expressed as dim (L n IP ) > 1 and means 3 implicitly that L is not constrained to lie in a proper subspace of IP .
More generally we can consider k-planes in TPn and their intersections
with subspaces of ]Pn. The variety parametrizing k-dimensional subspaces
n of IP is called the Grassmannian and is denoted by G^( 3Pn). Any condition
imposed classically can be formulated in the following way. There is a
strictly-increasing, nested sequence,
Ao C Al C ' " *C \ 5 1,11
of linear subspaces of JPn such that any. k-plane X satisfying the imposed
condition also satisfies
(Sch) dim (XnA.) > i 0 < i < k
and vice versa. So a geometric condition on a k-plane in IPn gives rise
to k+1 independent algebraic conditions. It could be shown that the i-th
condition is r.-"-fold where r^ = (n-k + i) - dim (A^) , but in practice all
k+1 conditions are considered together. Instead we show later that (Sch) is
an r-fold 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) is the dimension of G (JPn). (Sch) is called a Schubert
condition, and the set of all k-planes satisfying this condition is called
a Schubert variety which we denote by n[AQ,A^,...,A^J. It is a variety because it satisfies extra linear equations in addition to the quadratic
ones defining the Grassmannian which is embedded in projective space of
'n+1 dimension - 1. The Schubert varieties are then intersections of the !k+l - 11 -
N Grassmannian with certain hyperplanes in IP .
So the set of lines in 3-space meeting is represented by the 3
Schubert variety ^[L^, IP ] and so the set.of lines meeting all four given
lines is represented by the variety
4 3 i=l 3
Now the parameter variety in this case is G^( IP ) which has only one
defining quadratic polynomial and thus is a quadric hypersurface in .
Consequently V is defined by one quadratic, and four linear equations. The
elimination is obviously easily carried out yielding a single homogeneous
quadratic polynomial in two homogeneous unknowns and therefore the number
of lines, in general, meeting four given lines in general position in
3-space is equal to the degree of this polynomial which is 2x1x1x1x1 = 2.
Our history so far has brought us up to 1930 and van der Waerden's
foundation of the Schubert calculus. With Ehresmann in 1934 andi;his paper
Sur la topologie de certains espaces homogenes the calculus was put onto an
even firmer foundation. He showed that the 2i homology group of the
>s Grassmannian with coefficients in TL is generated freely by the classes of
Schubert varieties whose complex dimension is i, (the odd dimensional
groups are all trivial). For this reason a Schubert variety is also \
referred to as a Schubert cycle. This is the first part of what is called
the basis theorem and at this point we interrupt the history.
In the first two chapters we describe inddetail the parameter variety
G^( IPn) and its Schubert subvarieties. We prove (both parts of) the
basis theorem and show how the second part, which is really Poincare duality,
along with two formulae due to Giambelli and Pieri put the Schubert calculus
on a rigorous foundation by affording a complete description of - 12 - * n H (G iW ),7Z) as a 2Z-algebra.
In the last chapter we return to our history, outline some of the work done since Ehresmann and discuss the limitations of this and other inter•
section theories that have been developed since then. The problem of multiplicities also occurs in the theory of singularities of mappings which we mention briefly, as well as a description of the singular locus of a
Schubert variety wherein we include some of our own observations on the matter.
The scope of Hilbert's fifteenth problem is enormous. A great many mathematicians have contributed to its partial solution and their collective
efforts have given birth to new branches of mathematics, many of which have
already born fruit. But there remain those parts that, in their
elusiveness, invite the conception of even newer theories. - 13 -
Chapter I.
THE GRASSMANNIAN
§1 The Naked Grassmannian
In studying geometrical objects that are "curved" one technique is to consider all the best "straight" approximations i.e. the tangent spaces, and the question arises as.to where to put all these. And so we are led to consider the set of all k-dimensional subspaces of the ambient space. This object, however, is so interesting in its own right that we give it a general foundation. For this purpose, let E^ be a vector space of dimension n, then the set of subspaces of E of dimension k is called the n Grassmannian and denoted G, (E ). k n Let X e G, (E ), then relative to some fixed basis for E any k n n J ordered basis for X gives rise to a k* n matrix over the ground field W which has rank k. We call this matrix the Stiefel matrix of the chosen basis, and the set of all such Stiefel matrices for all X e G, (E ) we call k n the Stiefel space of k-frames in E^, and denote it by St(k,n). There is an action of the linear group GL(k, W) on St(k,n) by left multiplication.
Since any orbit of this action is exactly the set of Stiefel matrices representing all the ordered bases of a given subspace in G^(E_), the
Grassmannian appears as this quotient.
Heuristically speaking, every point on the Grassmannian looks like any other point. Strictly speaking there is an action of Aut(E^) on E^ that induces an action on the k-subspaces of E^ which is transitive.
Relative to a fixed basis for E we have Aut(E ) = GL(n, IF) and a n n decomposition E^ = E' $ E" where E' is the span of the first k basis vectors and E" is the span of the remaining (n-k). If two automorphisms both take E' to X e G. .(E ) then their ratio leaves E' invariant, thus k n •
G^CE^) can be relabeled as the space of left cosets of the isotropy group of E' i.e. GL(n, ]F)/Isot(E').
These two points of view are reconciled as follows: a matrix
A e GL(n, IF) sends E' to the subspace of E^ spanned by the first k columns of A, so let us write A = (A^l*) and define the map
m: GL(n, TF) — St(k,n)
t via
where Afc denotes the transpose of A^. This map is clearly surjective.
Consider the diagram
m GL(n, IF)c *» St(k,n)
(1.1.1)
G. (E ) k n
where ¥ is the projection defined above and
If (A^|*) g GL(n, W) then
We note in passing that IsotXE') consists of all matrices of the fo
N rl-k - 15 -
where 1^ e GL(k, IF), ^n_^ e GL(.n-k, IF), so Isot(.E') Is usually written
GL(k,n-k, IF) .
All that we have dealt with so far is the Grassmannian in the light
of "just linear algebra," and though we will continue to get more mileage
from this, we are specifically interested in examining structures that are
derived when has a geometrical.foundation. In differential geometry
then, we use the ambient spaces Hn and (En and the corresponding G ( IRn)
and G, ( of the geometric structure that we examine is purely algebraic and for this reason we avoid the hassles of non. algebraically closed fields. Henceforth then Ei will be the affine space of dimension n over IF, which we assume to be algebraically closed, and so in the particular case where IF = (E we are dealing with, two topologies on E^., the Zariski topology and the usual topology. With either topology on E^ = IFn the projection from the n-fold product of IFn with itself onto the first k factors is a continuous open N2 mapping. GL(n, IF) is topologically. IF - R, where R is the zero set of the determinant function therefore closed: in both topologies. On the other kn hand we have St(k,n) is topologically IF -S, where S is the set of matrices with rank strictly less than k, but such a matrix is characterized by all its k x k submatrices having zero determinant, thus S is also closed in both topologies. Certainly S is contained in 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 in diagram (1.1.1). Since R and S are closed, m is also open and continuous. By the preceding argument and the - 16 - commutativity of (1.1;1) we deduce that Y and We conclude this section with the observation that when k = 1 the Grassmannian is exactly P(E ), the projective space associated to En , showing that ^(E^) is a generalization of one of the most important concepts of geometry. §2 The Grassmannian as Variety A great deal of the structure present in the Grassmannian is appreciated by seeing how it presents itself as an algebraic variety. As we have seen, a special case of the Grassmannian is projective space, so one would not be surprised to find out that the Grassmannians are all projective varieties. To get an algebraic hold on the points of G,(E ) the exterior powers K. n k come to hand easily. If X is a k-subspace of En> then A X represents a "line through origin" in the vector space A E . Now, a basis n lk k, ' k k for X if and only if X^A...AX^ generates A A, so if A X = A X' then X = X' thus we get a canonical mapping. p : G, (E ) + !P(AkE ) k n n which is injective. We show that the image is closed with respect to the Zariski topology on 3P-(A^E ) ; and thus G, (E ) clothes itself as a projective variety. To n k n J see this choose a basis for E say {e,,...,e } then e. A...Ae. : n 1 n i, l' 1 k k k 1 < i, < . . . < i, < h is a basis for A E , so the points of TP (A E ) are k k n n represented by their homogeneous coordinates (...,x. . ,,..) relative to i1...ik k n this basis, and an affine open cover of IP (A E ) is given by the (, ) sets n K. 17 - U . of points with homogeneous coordinate x. . ^0. By elementary 1 ,,,m1 ir....-,.ik 1 "k topology it suffices to prove that p(G (E )) nU. . is Zariski closed in U. Ll' Without loss of generality assume i. = j, and set U „ = U. Let 3 X , Z , . . . , K e and x the s an of E' be the span of e1>...,e , E" the span of \+i>•••> n P x ^,...,x^, then each x_^ has a unique representation e^ + e\| where e^e E', eVe E". So x X;LA...Axk = + e^ACe^ + e^)A...A(.e^. + e^') = e'Ae'A...Ae' + (terms that are zero, or else not in Thus p(.X) eU iff e^A...Ae^ = Xe^A.-.Ae^ for some \ 4 0, but then e^,...,e£ is a basis for E' showing that we could have chosen the e_^'s originally to give X a unique basis of the form e^ + w^, e^+w^,..., e^ + w^ where w. eE", This is of course tantamount to choosing the affine coordinates u. = x. . /x .We have J -|_»" " *' k ^l'*""'^k 1>2,...,K „ ,s X,A...AX, = e1A...Ae/+ Y e A. . . AW. A.. . Ae, 1 1 fc (1.2.1) 1 k 1 -.k i + V e, A. . . AW. A. . . AW. A. . . Ae, + . . . + W, A . . . AW. •* , , , 1 x i k 1 k l that p(G,(E )) nU is parametrized by the w.'s. These w.'s in turn will determine a kx (n-k) matrix B relative to the basis e. e , in fact k+1 n B = B is the Stiefel matrix of x^,....,x^ is ( i kl' ) determined - 18 - completely from the second term in (1.2,1) by (1.2.2) u « = (.-l)k_:L a... 1 < i < k l,z,...,i,...,k,j ij k+1 < j < n, where the circumflex over a subscript means that the subscript is taken out. This has the effect of showing that p(G,(E )) is covered by open sets each canonically isomorphic to affine space of dimension k(n-k) and the rest of the affine coordinates are clearly related by polynomials to those of (1.2.2). Though some might be content to stop here, we shall press on to find these relations explicitly. First let us note that the alternating k k-linear form defined on X by the equation i=l P. (X) = P. (x ...,x ) = u. Jl»,,,,:,k 3l'',-'Jk 2V',2k is independent of the coefficients of all the e.'s, save e. ,...,e. , in the expansion of the x_/s and so it is actually a function of the rows of the kxk submatrix of the Stiefel matrix formed by taking the J-^* • • •»jj^1 columns. As such P. . must be a non-zero, scalar multiple of the Jl',,,,Jk determinant function; but this scalar is clearly independent of the rows chosen since P., ., will perform, exactly the same sequence of arithmetic Jl"*-,:ik operations on columns j ' ,.. . j ' to obtain x., ,, as P. . does on 1 k 3l'"""'~'k its columns to obtain x. . . Denoting this kxk matrix by A. . , 2 3 : J1>-.-»Jk V'-> k it follows from the observation of P^ ^ ^(x^,...,x^) = 1 that - 19 - (1.2.3) u. . = det(A. . ) 21'2 2' * * *'2 k ~'l'*'"'~'k Expanding by minors along the row gives us (1.2.4) u. . . = J ("Da+1a . detCA?'1 . ) 2l'22'"'2k 1=1 aJi 2V",2k 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 i1"*1 column. The a*"*1 column of A = 1L is all zeroes except for a 1 in the row, so clearly we have, on replacing the i^ column of A. .by the column of A1 „ , , jj>...»Jk X,Z,...,K (1.2.5) detiA®'1 . ) = det(Aa,i . - : 2 V"3k ll"-" !^, A' where the circumflex beneath the subscript means that it replaces the one taken out. Furthermore replacing the a*"*1 column of A. „ , by the i^ ij Zj • • • j K column of A. .we obtain, as in equation (1.2.2) Jl5--..Jk (.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^,^,-...jj^ or in homogeneous coordinates - 20 (1.2.7) 4- " x ,x. . - Y (-1)06 Xx „ . x. A . = 0 X , . . . , K. Ji.''"*'~'k i=l x,. .. j0tjj^j...jl£ J 2> • • '»J >• » • 'Jj^ where it is understood that the x. . are alternating in their indices. Thus we have shown that for any Xe p(G, (E )) n U there is a point in IP (A E ) k n n whose homogeneous coordinates satisfy (1.2.7) for 1 < a < k, 0 < J1 < J2 < • • • < Jk < n. Conversely we show that any point (...,x. . , ...) e IP(A E ) x^ > • • • > ^k satisfying x^ ^ t u and equations (1.2.7) is indeed a point in p(G (E )) n U. Without loss of generality we may assume that x , = 1. kn X , • • • , K. Define a kx n matrix (a^j) (as i-n equation (1.2.2)) aii = Xl x ki' But if 1 < i < k then a.. =6.., the Kronecker delta, thus (a..) is the ij ij ' xj Stiefel Matrix ( |B) of a k-dimensional subspace X of E such that = pn „ u(^) !• Now consider the matrix formed from by replacing one of its columns by a column from B, then the determinant of this matrix, k-i i.e. p. <» , . (x) is simply (-1) a.., but since the rest 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",3k Jl'**"'Jk for all sequences j^,...,jk« We summarize the above discussion in the following theorem: - 21 - Theorem (.1.2.8) The mapping p : G, (E ) >-IP.(.A E ) is a closed embedding, giving K n XI G^(En) the structure of a non-singular projective variety of dimension k(n-k). The P. . (X) are called the Plucker.coordinates of X and Jl'"*,J.k p is called the Plucker embedding. This theorem allows us to make precise the notion of the Grassmannian as a parameter variety. When we say that G^(E ) parametrizes k-planes in n-space we mean that there is a one-to-one correspondence between the set of k-dimensional subspaces of E^ and the projective variety p(G, (E )). Henceforth there is no need to distinguish between k n G, (E ) and its image under the Plucker embedding and so.we identify the two. A point X £ G^(E^) is the solution space to a system of homogeneous linear equations with rank (n-k). Since this system equally well describes conditions on the homogeneous coordinates of ^"(En) yielding the projective linear subspace 1P(X), G (E ) may also be thought of as parametrizing the ic n (k-1)-dimensional projective linear subspaces of projective (n-l)-space. We write BG (p(E ))BG {Fn 1(I)) W k-i fi k-i ' - The simplest example of a Grassmannian which is not a projective space is G^CE^) . In the case where IF = (E, this is the same as the space 3 G,( IP ) which is mentioned in the introduction. G„(E.) has dimension 4 1 2 4 and so it is a quadric hypersurface in 3P^( IF) having the single defining equation X12X34 X13X24 + X14X23 " °' - 22 - In the next section we view the Grassmannian as a complex manifold. If IF = IR, G, (E ) is. a real manifold also, but we do not discuss this for k n reasons, mentioned before. In either case, however, the Grassmannian is compact, being a closed subset of projective space which is compact. §3 The Grassmannian as Manifold The results 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 establish that G, (E ) is covered by open sets k n J W. = G (E ) n U. . which are all canonically isomorphic as J^>"'-»Jk k n -'l'*""'^k affine spaces of dimension k(n-k). These isomorphisms, in the case where ]F = (C, are also biholomorphic and so G, (E ) is the complex manifold associated to the algebraic variety of Section 2. We can see these charts arriving in a slightly different way from kn St(k,n), which, now being an open subset of (C with the usual topology, takes its rightful place among the manifolds. As in Section 2, for any Stiefel matrix A let A. . be the matrix of columns .j. j then J 2»• • •»Jk k the set V. . = {AeSt(k,n); det(A. .WO} Jl' •' • ,Jk Jl',,,,Jk is a Zariski open subset of the Stiefel manifold which is evidently stable under the action of GL(k,(C) and so the image cp CV. . ) is an open Jl,,-,,Jk set in G (E ) which is, of course, the set W. . above, k n 3 1' * * ' '^k As we have remarked in Section 1, the Grassmannian is the same all over. The group GL(n,(C) acts transitively by automorphisms which are linear, whence algebraic and holomorphic, and so G^(E^) earns the title of a homogeneous space. This fact can also be seen in a slightly different way when we think of (Cn as endowed with its usual hermitian inner product. The transitive action is now given by the unitary group U(n), and the isotropy of E' is denoted U(k,n^k). Consider first the continuous map t : GL(n,(C) > GL(n,C) via A > AA A where A denotes the conjugate transpose of A. U(n) is the inverse image under t of the closed set consisting of the identity t^, and so is closed itself. From the equation AA = H for A = (z..) eU(n) we have, n ij n z . . = z . . z . . ^ V z . . z . . = 1 showing that U(n) is both closed and bounded, i.e. compact. Since G, (E ) k n can be identified as U(n)/U(k,n-k) this point of of view has the advantage of showing that the Grassmannian is compact, without venturing into the algebraic category. We would, however, have ended up naturally at projective space anyway, since interesting compact, complex manifolds can't live in an affine environment. It would be unwise to continue to separate the discussion into distinct categories since part of the charm of the Grassmannian is how the various structures flow into each other. Henceforth then we shall assume tacitly all 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 itself is a natural choice. We must now demonstrate the existence of local trivializations and show the compatability of these on the intersections by explicitly defining the transition functions. To this end recall the map 0 : St(k,n) -> G (E ) as defined in section 1 K. n and the open cover {W^.} as defined in section 3 where I = (i , ...,i ). It is easy to see that 3> is holomorphic in this case. Denote by U the set K. consisting of all pairs.(X,x) where Xe G. (E ) and xe X, so that k n U, c G, (E -) x Cn. Define k - k n JJ : U, G. (E ) k k n via (X,x) -> X -1 k and we must first exhibit homeomorphisms F^. : IT (W^.) W^x (£ } i.e. we wish k -1 to use (E as a canonical model of each of the fibres TT (X) for each XeWj. Now, for a given X e W a Stiefel matrix A associated to X has the property that the kxk submatrix A^ formed by taking the i-j^' ^2^' '^k^ columns is non-singular, and we can without loss of generality assume A^. = 3J^. A vector x e X is a linear combination of the rows of A, and the coordinates of a vector Ve : (X,x) -> (X>v) is then the required one. On the overlap Wj. n Wj, X is represented by a matrix A such that A^. = ll^ and such that A is non-singular, where A is the kxk submatrix formed by taking columns j^,...,Jk- The matrix A, representing X and having A^. = ll ^ is - 25 - unique, likewise the matrix A- representing X and having A^ =. i . thus A = T A' where T = A e GL(_k, C) is unique. The homeomorphism X X J F F"1 : W n W x ck -> W n W x is given by (X,x) —• (X,T (x)) and the mapping tIJ : wTn WJ GL(k'c) given by X —>• T is evidently holomorphic. Thus U is 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: let a k k n k S T : WT WT x be defined by S T(X) = (X,C ), where A is as before and C is the a, i- a a column of A. S is clearly holomorphic and S. ,...,S. generate each a, I 1^,1 """k fibre over W^.. It remains to show that this way of defining a section is truly global, i.e. that it is compatible with the transition functions on 1 1 overlaps. If XeWT also, then S T(X) = (X,C') where C' is the a "* column J a,J a a of A', and A' is as before. But we see immediately that C = A C' a J a which is all that is required for the patching. The following theorem justifies the usage of the adjective "universal" when referring to U^. Theorem: Let M be a.complex manifold of dimension n. If K-> M is a holomorphic vector bundle rank k, generated by n global sections r^,-. i .-,-r , then there is a holomorphic map - 26 - The universal bundle U, is a subbundle of the trivial bundle G, (E )x (Cn. k k n We denote the quotient bundle by Q ^ which has rank n-k and which is called the universal quotient bundle on G, (E ). Thus the sequence i£ XT 0 -»• U, G, (E ) x Cn Q , -* 0 k k n n-k of bundles over G, (E ) is exact, k n §5 The Dual Grassmannian If we consider for a moment a k-dimensional linear subspace X of 3Rn we see that there is a unique (n-k)-dimensional subspace corresponding to X i.e. the orthogonal complement X^ relative to the usual orthonormal basis. An isomorphism G, ( IRn) ^ G , ( 3Rn) is then obvious, but this does k n-k not work for an arbitrary ground field and depends on a choice of basis. The idea that the set of k-planes in n-space'should look like the set of- (n-k)r-planes in n-space can be formulated naturally as follows: let E = Horn ^(E , IF), where E is a vector space of dimension n over an n JF n • n arbitrary ground field 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 is easily 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 = {AeE : d>(f) =0 f e Y} and where n is o on A the canonical isomorphism. G , (E ) is called the dual Grassmannian. n-k n In the case where IF = n o+.u , -»• G . (E ) x e ^ q, 4 o n-k n-k n k _1 * ~ ~ and Q • = (d ) (U, ) where U, is the dual bundle of U, . k k k k - 28 - Chapter II THE SCHUBERT VARIETIES §1 The Definition We have up to now considered the Grassmannian as a completed form, but its true fascination lies inside. The alignment of the kr-dimensional sub- spaces with each other provides a means of classifying them even though each one was previously undistinguished by virtue of homogeneity. To examine this alignment we consider the filtration on E^ determined by the chosen basis e, e , that is 1 n (2.1.1) 0=EcE,c...cE o 1 n where E. is the span of e,-, ...,e.. We can think of E, as having the "best l 1 x k ° alignment" with this filtration 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 this end then we consider n \ which we call the intersection the sequence of integers j^dim (X n 1=0 sequence of X and denote by i(.X) . In particular we have i(E ) =. (0,1,2,3,...,k,k,k,...,k) and this is our basic sequence. i(X) is always a non decreasing sequence starting at zero and becoming eventually constant with value k. At each stage we allow one more dimension expansion, so intuitively we should expect jumps in the sequence of height at most one. This is seen to be true by inspecting the pair of exact sequences i+1 0 —• X n E. —• X n E. ,, -4-x —T+ JF i x+1 0 —> X n E —• x n E —XnE_/XnE. —> 0 X X+X X+l X - 29 - where x^"*" is the projection onto the (i+l)st coordinate; that is Xn E.,,/Xn E. has dimension 1 or 0 depending on whether or not Xn 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 intersection sequence then, looks like C-0 J 0 J • • * jOyXj-Xya • • j X « 2 • 2 ^ • • • y 2 j • • • y X y lt—1 y a • • y lC~* X j lC j lC j a a a y k.) where the zeroth place is always zero. Comparing i(X) to i(E ) we. see that.the difference is in the place where the dimension jumps for the i^. time. For i(E^) the 1^ jump occurs at the th i place, but in general there is 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, in turn, are uniquely determined by i(X). A property of i(X) is that each integer 1 < i < k appears at least once, but the number of times it does appear is exactly (a^+^ + i+'l) - (a + i) and thus a^+i~ a± ~ ®' Thus we have a bijective correspondence between the set of intersection 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 bijection between JJ and JK, the set of strictly increasing sequences 1 < a| < a^ < ••• ^ a^ < n, so we see immediately that r \ n the number of distinct intersection sequences is We can think of the sequence (a a ) as a measurement, in some sense, of how awkwardly the k-plane X sits relative to the chosen filtration on E. But we can also, use (a^,.,.ya^) as a bound on far we allow this awkwardness to range as we vary the k-plane X, So we consider the set of k-planes whose th intersection sequence has a .lag of at most a^ in the position of the i jump. Let us denote this 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. .by the following: Xl'•••'1k Proposition (2.1.2) fi(a^,...,ak) is the subvariety of ^^(E^) corresponding to the linear polynomials x. . where j , . .., j is any sequence such that j > a + X 1' " " k . A for some 1 <_ A <^ k. Proof: Let Xefl(a^,...,a^) and let j > a^ >-'X for some 1 <_ \ <_ k. Since dim(Xn Ea +_^) >_ i, 1 <_ i <_ k we may. choose a basis, x^, . . . such that i x.e E , ., so the Stiefel matrix of x, ,...,x, looks like x a.+x' 1' k x X X X 00 0 11 12 ••' l,ai+1 X X x x 00 0 21 22 2,a+l ••• 2£2+2 XA1XX2 ••• XA,a£fl *X^+2 +\ 00 x, „x X X 00 ... 0 kl^k2 '•• k,a;L+l k'a2+2 "k.a^+X "k^+k consequently P. .. (X) is the determinant of the matrix 31 - 1 o x... X'\*-l X XA+1 A+I,j1 * * ' ^A-I x. . x . . . X . Xk J k,3x ^•>31 ' A-l k,Jk Using the Laplace expansion of the determinant we get directly that P. . (X) = 0, thus the linear polynomials all vanish on fi(a ,...,a ) Conversely, consider a point X of G, (E ) whose Plucker coordinates k n satisfy the linear relations. We pass to the affine coordinates on W where ...£ is the sequence, chosen from among those for which -£•1 9***9 <£i_ k x ^ 0, which maximizes the sum J. j . From §2, Chapter I we know J i > •J • • »Ji ~r 'V"' k r=l that the point with Plucker coordinates x. has a basis x^,...,x .k-i whose Stiefel matrix is (x. .) = ((-1) u„ ). Since u,, p t 0 then £. < a.+i. Now for any j > a.+i we have 1'"' * 'vi — k k r=l r=l r^i so by the maximality of Y., £ we get x. . =0. This shows that x sE r=l r 13 1 ai + : In the same way we get for i' < i - 32 - l i thus dim (Xn E& +^) > i putting X in f2(a . ,.>a ) q.e.d. i Note that fi (a, > • • • »'a. ) e. G (E ) corresponds to the variety Q[A , ...,A. , ]CG, , (1P(E )) as defined in the introduction where A. = P(E ,.) o Tc-1 - k-1 n I a.+x l and so we are completely justified in calling ft(a ,...,a^) a Schubert variety. The condition dim (Xn E, , .) > i 1 < i < k a.+i x is called a Schubert condition, which is also consistent with the definition in the introduction. Finally, for convenience, we call (a ,...,a ) a Schubert X K. symbol over G, (E ) . k n Some examples are in order here. (2.1.3) n(0,0,...,0) In this case the k-planes are not allowed to roam at all, and so this variety must consist of just E^. To.see this properly note that, in particular, 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 is d, 1 < d < k For Xe fi(0, .. . ,0, a, • • •, a) again in particular dim (X n E^^) ^ k-d and so c c.X, Similarly since dim (X n E^) > k we have x E^+k. Conversely suppose E, , cgcE . 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 satisfies the requirements for lying in £2(0,. .., 0, a,.. ., a) , thus - 33 - n(P,...,0,a,...,a) = {Xe GkCEn) ; E^cXc E^} and by sending X to X/Ek_^ we see that £2 (0, . . , ,0,a, , . • ,a). is isomorphic to VW Vd} = VVd) • C2.1-.5). £2(n-k,n-k,. .. ,n-k) This variety allows the largest possible lags, consequently the largest Schubert variety. It is in fact the whole of G, (E ), since for 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 certainly 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 Varieties in G2(E^) We now illustrate the previous section by attempting to visualize the geometry of the Schubert subvarieties of ^2^r? ' This Grassmannian, as we have mentioned before is a quadric hypersurface in TP'* with the single defining equation X12X34"X13X24 +X14X23 = °- The Schubert varieties 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 - 1 52(0,1) = G1CE2) = JP , 2 9.(0,2) £ G3(K3) s JP , G (E ) G (E ) = fiU.l) £ 2 3 = 1 3 and from example (2.1.5) we have that £2(2,2) = G2(E4). The remaining Schubert variety £2(1,2),. is more difficult to describe due to the fact that it is not smooth. It is the smallest example of a singular Schubert variety, and the remainder of this section is devoted to its description. In addition to the quadratic relation above, (2.1.2) tells us that, on $2(1,2), we have the relation x^^-= 0, that is J2(l,2) = V(x12x34 - x13x24 + x14x23) n V(x34) = V(x14x23 - x13x24) n V(x34). 5 3 Note that TP n V(x .) n V(x ) = JP where the homogeneous coordinates 9 1 » ^ x x x and X and so n x 2 v x = V X X X X nV X are 13' 14> 23 24 ( > ) n ( 12) ^ 14 23— 13 24^ ^ 34^ 3 n V(x.„) is a subvariety of JP . Since x,,x.„ - x,„x„, does not involve 12 14 23 13 24 x_. or.x,„ we can write this variety V'(x.,x„„ - x,„x„.). This is simply the 34 12 14 23 13 24 11^ 3 image of JP x JP under the Segre embedding into our choice.of JP thus we have 1 1 (2.2.1) £2(1,2) n V0xl2) s IP x JP 5 On the other hand £2(1,2) n {X e JP ; x12 f 0} = £2(1,2) is a neighbourhood of (1,0,0,0,0,0) in £2(1,2) and we can look at it in terms of the affine coordinates on W^2. Recall here the definition.of the standard - 35 - open cover of G^CE ) as defined in Chapter I, sections 2 and 3, and set w = fi(.l,2)n W . For Xg W the Stiefel matrix A, which represents X, such that A = i 12 2 is given by equation (.1.2.2) as fl 0 -x23 -x24l A = ° 1 X13 X14 X X X X = where x^^ = ^3 24 ~ ]_4 23 ^" Since x^2 = det A^2 = 1 we can consider X13 'X24'X14 anc^ X23 aS a^ine coordinates and as such is a quadric 4 hypersurface in C . The dimension of the Zariski tangent space at the origin is 4, but everywhere else it is 3 showing that (1,0,0,0,0,0) is an isolated singular point of £2(1,2). By a similar analysis on each of the other open sets we can see that this is, in fact, the only singular point, thus (2.2.2) Sing £2(1,2) = £2(0,0) . For the remainder of':this discussion, then, we fix the ambient space as 4 5 JP = IP n V(x^4) . We can now view £2(1,2) as the singular affine variety w^2> completed by it points at infinity, i.e. those lying in the hyperplane x^2 = 0. As we have seen, the restriction of the Plucker embedding to this set of points at infinity is the same as the Segre embedding of JPXx JPX into 3 4 JP = JP n V(x^) . Since £2(1,2) is covered by the sets , open in £2(1,2), and since JPXx JPX n = 0 it follows that 1 1 (2.2.3) JP x JP = U w..nV(xl9). i=l,2 13 j=3,4 - 36 - This last union can he broken down more usefully as follows.: (2.2.4) ._V -w±j = W13 * [W14X W13] U [W23 XW13J 11 [w24X (w14UW23)J x x, z j=3,4 where \ denotes the set difference. Relabel the sets on the right ;as D D and D D1, 2' 3 4 from left to right. To show that 4 i=l is, in fact a disjoint union it suffices to show that Dxn D4 = D2n D3 = 0 but a point in D1 n must have = 0 = x^ and x^2 ^ 0, and a point in D2 n D3 must have x^ £ 0 =f x^ and x^ = 0. Both of these are impossible x x since x^x^ = 13 24' Consider P"^ = C XL'{:OO} , where oo is the point at infinity then we have IP1 x TP1 = (.C x C) XL '(W'x C) XL (C x {«>}) ii { (oo,*,)} . Again we relabel the sets on the right hand side as C, , -.C_, C_ and C, in 12 3 4 the order that they appear. We wish to show that there exists such a decomposition satisfying (2.2.5) C.cD. for i = 1,2,3,4. xx • ? ? » First we choose local coordinates on each w. . for i = 1,2 and j =. 3,4. Let ij ij xi A be the Stiefel matrix of Xe w. . such that the 2x 2 submatrix A., of columns i and j is the identity, then, by analogy with equation (1.2.2) we - 37 - have,: '1 V 0 '1 V 0 0 ' 13 l U 2 A = A = Q Z 1 u 0 Z U 1 l l 2 2 f U 1 0 o • 1 0 0 ' 23 3 24 A = A = Z 0 1 U Z 0 V 1 3 3 4 4 X X X X X ) where, setting v = ( 12' 13' 14' 23' 24 ' v = (z x u v u x 13 i > r r iV' 1 v = (z2, u2, 1, u2v2, v2), A14 1 v = ( z U U V 1 V } X " 3> 3' 3 3' » 3 ' 23 1 v = (_z U V U V 1} X 4' 4 4' 4' 4' 24 Since the affine coordinates u,v and z on w„ are allowed to vary freely we see immediately that each w.. is an affine space of dimension 3. In particular we have dimc (fl(l,2)) = 3. Restricting the local coordinates to w!. = w.. .n V(x ) reveals how (2.2.5 ) ij ij 12 is satisfied. From (2.2.3) and (2.2.4) it follows that 1 1 IP x W = w|3 IL [w]_4 \wj_3J JJ, [w23\w|3J ji {w\fi \ (w]A uw;,)J "24 x v"14 u"23' D!, where D! = D.n V(x ). . n x xx 12 nx=l - 38 - Setting x = 0 amounts to killing the z-coordinate on each w.., so we have Ll IJ ' ?i; v1 0 0 D! = ; u v e c > = c-x c. x 0 0 1 u In addition, setting x = 0 amounts to killing the u-coordinate on w', and XJ 14 W23' so we axso nave 1 v2 0 0 {co} D c CX 2 ; v2 e 0 0 0 1 and 0 10 0 ; v3£C = WxC, 0 0 1 v. Finally, setting = x23 = 0 amounts to killing both the u and v coordinates on w^., and so the last set is 0 1 o o| D4 = \ = {(»,»)} 0 0 0 1 D! then is the required set C. and we have established (2.2.5). - 39 *- Chapter III THE SCHUBERT CALCULUS § 1 Intersection Theory In this section we summarize briefly the main ideas in the topological intersection theory developed by Lefschetz, At the time of writing, the book Principles of Algebraic Geometry by Griffiths and Harris has been recently published. This book contains a complete, up-to-date version of this theory, so any detailed treatment here would be redundant. oo Throughout this section M will be a real, oriented C manifold of dimension m. A singular p-chain C = on M, satisfying the property that each singular p-simplex, r, , is the restriction to the standard p CO p-simplex A c JR of a C! map from a neighbourhood of A to M is called P P a piecewise smooth p-chain on M. Since the boundary of a piecewise smooth D S chain is piecewise smooth, we can define a chain complex C^ (M,Z) which is a subcomplex of the singular chain complex. It is a fact from differential topology that the homologies of these two chain complexes are isomorphic. An (m-p)-cycle and an (m-q)-cycle are said to intersect properly if the intersection has pure codimension p+q. They are said to intersect transversely at a point x if the tangent space to the intersection at x has codimension p+q also. Let A and B be two piecewise smooth cycles on M of complementary dimensions, i.e. dim A = p, dim B = m-p and let x e An B be a point where JR JR A and B intersect transversely, i,ev, 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 respectively. In fact - .40 - T CM) = T (A) f T (B) , XXX Let {u1?,,,,u } and {v.,,,,,v } be oriented bases for T (A) and T (B) 1 p 1 7 m-p x x respectively then we define the intersection index of A with B at x as follows det (u ,., . , ,u ,v ,,. , ,v ) i (A,B) = 1 . • • E_JL _JtP__ det (u , , ,, ,u ,v ,,.,,v ) 1 p 1 m-p aat is, + 1 according to whether or not {u.,...,u ,v.,.,,,v } is an J- p 1 m^-m«-pD oriented basis for (M). In the case where A and B intersect transversely everywhere we define and call it the intersection number of A with B. The word number is justified since, by hypothesis, dim (An B) = 0 so AnB is a discrete subset of M, which is assumed to be compact. .Thus AnB is finite. One shows that the intersection number depends only on the homology classes of A and B, that is if A is homologous to A' then (3.1.1) For any two homology classes a e H (M, 2Z) and g s H (M, 2Z) it is possible ^ P m-p to find two piecewise smooth cycles A and B representing a and 3 respectively - 41 - such that A and B meet transversely on A nB, So we can define a bilinear pairing on homology H (M, TL) x H (M, TL) + TL p m-p ' via (a,S) + {a, B.) = (A,B), which is called the intersection pairing of a with 3. The Poincare duality theorem asserts that this pairing is unimodular, in other words that Horn™ (H (M, TL) , TL) ={{a, ) ; a e H (M, TL)} , iL m-p P H M we and also that if, for a fixed et. e P( J %•) , have (a,g) = 0 for all B s H (M, TL) then a is a torsion class. In r particular if H (M, TL) is free m-p p then is non-degenerate. In the course of proving Poincare duality we find that there are isomorphisms m_p D : Hp(M,Q) -y H (M,4)) satisfying (3.1.2) (a.&) = [3,D(cO] ^jhere [ , ] denotes the Kronecker product. We will find (3.1.2) useful later. Q is used as the coefficient ring simply to kill the torsion. In view of the next section this is purely a technicality, We are now in a position to present the main points of this section. To this end let M now be a compact, complex manifold of dimension n over m = 2n. Let V be an analytic subvariety of M of complex dimension 6, and set p = 2 3. Let V have its natural orientation. We can assign a cohomology class - 42 - to V in the following way. Let a e p^s> ^ and choose a representative cycle A that meets V transversely in smooth points, Again one shows that the intersection number is independent of the choice of representative A for a, but this is not as straightforward as in the case of (3.1.1) due to the possibility of singularities on V, We get around this problem by using the following trick; suppose A and A1 are both representatives of a, then, using the fact that the singular locus of V is a proper subvariety of V, hence has real codimension > 2, it is 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 in the case of (3.1.1). For this, the interested reader is referred to Griffiths' and Harris' book. V then defines a linear functional H (M, ZZ) -+TL, which by m-p ' J Poincare' duality is of the form } for some yeH^(M, ZZ.) . This class or its Poincare dual D(y) g Hm ^(M, ZZ) is called the fundamental class of V. One might ask if all this is 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 this, in general, is not possible. It is proved in the paper Nonsmoothing of algebraic cycles on Grassmannian varieties by Hartshorne, Rees and Thomas that the Schubert variety £2 (.2,2,3) in GQ(E^.) is not "topologically smoothable," - 43 - Suppose W is an analytic subvariety of M having complex dimension n ^9 and intersecting V transversely at x, Since there is a natural choice for the orientations of V, W and M it follows easily that l (.V,W) = 1, x This fact is central to algebraic geometry. It reflects a basic difference between real and complex geometry, If V and W intersect transversely in a finite set of points then the intersection number is the number of points in the set-theoretic intersection, whereas in the real case a cycle may cross another cycle, change direction and cross back producing a cancellation. So, in the real case, the intersection number is usually somewhat smaller than the cardinality of the intersection. Now suppose that V and W do not intersect transversely. Essentially this means that some tangent directions to V and W at a point xeVnW have coalesced. By perturbing W slightly inside its homology class we can count the lost tangents. This is done locally by finding an analytic variety of dimension m whichiis a u-sheeted branched cover of a neighbourhood of x whose fiber over e is the intersection Vn W + e. Here we think of n £ eC 3 v,W. mx(V,W) = u is called the intersection multiplicity of V and W at x,. and does not depend on the choice of local coordinates, If V and W intersect in a finite number of points then, by an explicit local analysis at each of the points x^, it is possible to find a representative W' of the homology class of W which meets V transversely in m (V,W) points in 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 intersection point is characterized by having unit multiplicity, It is a simple matter now to see that the intersection pairing is Poincare dual to cup product in cohomology, Consider the following diagram Hm P(M,U)) x HP(M,q) ([M]n-) x id P K Hp(M, id. x D J H (M,OJ) x H (M.Q)- p m-p where D is the Poincare duality isomorphism, I the intersection pairing and K the Kronecker product. Commutativity of the lower triangle follows from (3.1.2) and the fact that for analytic cycles veH (M, ZZ) , weH (M, ZZ) J P m-P 1-X(V,W) = lx(w,v) = +1 consequently (v,w) = (w,v). [M] is the fundamental class of M and n denotes the cap product so [M] n- i an alternative description of the Poincare duality isomorphism. Since cup and cap products are adjoint with respect to the Kronecker product , commutativity of the upper triangle follows where C denotes the cup product. - 45 - In case ex,(3 are two homology classes not of complementary dimensipn we can still define an intersection product. Suppose aeH (M, 2Z) and • m-p g e (M, ZZ) , there exist representatives A and B intersecting transversely almost everywhere, We choose the orientation for C = AnB so that if {v..,...,V } is an oriented basis for T (C) on smooth points x, 1 m-p-q x and if {Ul""'VVl""'Vp-q} and {v V W W } l""' m-p-q' l'"" p are bases for T (A) and T (B) respectively then {U1" ' ' 'Uq'V • • "Vm-p-q'Wl' ' ' * 'Wp} is an oriented basis for T (M). C is then called the intersection cycle and we denote it A-B, One checks that this product is well-defined on homology by finding a chain D intersecting B transversely almost everywhere such that 3D = A and then showing that 3(D * B) = A • B holds when Orientations are taken into account. One also checks that the product is associative, This product and indeed allcof the preceeding is central to the justification of the Schubert calculus. - 46 - §2 The Grassmannian as C.W, Complex The application of an intersection theory to the subvarieties of the Grassmannian would be greatly^enhanced by an explicit basis for the homology. This was first described by Ehresmann who showed that the Schubert varieties (or more precisely their interiors) provide a cell decomposition of ^(E^). Recall the discussion in the first section of chapter one. There we saw that every k-plane X has a unique intersection sequence i(X) which in turn corresponds to a unique Schubert symbol, that is the sequence (a^, . . . ,a^_) of lags in the intersection sequence. Therefore G. (E ) is decomposed as a rC XI disjoint union of the sets (a^,...,a^) where u(a^,...,ak) = {XeG^XE^) ; (a^,...,an) is the lag 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 is the complement in ft(a^,....a^) of the union of all its proper Schubert subvarieties. By choosing these subvarieties as large as possible inside n(a^,...,a^)- we can show the following; Proposition (3.2.1) afc) = 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 If b. < a. and dim (XnE, ,.) > i then dim (XnE ,.) > i, thus if this ii D .+i a ,+i I I holds for 1 < i < k then ft(b. , . ,, ,b. ) c ft(a. , , ,, ,a, ) . But if b. < a. then the 1 k — 1' k ii decomposable k-yector A en A e A . . . A e . , , \ e, , . , N A . ,. A e , . A , ,, A e 1 2 i-(a.-b.) b.+i+l a.+i a.+k iii I l - 47 - satisfies dim (X n E , .) =? i but not dim (Xn E, , .) > i, It follows that a. TI b . ti l l fi(b1? , , ,? bk) c. fi(a1?, , , ,ak)< = >bi < a±, 1 < i < k and that the containment is proper exactly when at least one of these inequalities is strict. Thus u(a ,...,a,) is non void, X K. Suppose X e oo(a , , ,, ,a ) then dim (X n E n) = i-1 < i thus X K. cl ,Tl ^X X £ tt(a^,, . , .a^a -1, . . .a^) = £2X for all i, Conversely if X k £21, for A all i, then dim (X nE& < 1. However it is a property of i(X) that it 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 for a moment the example of £2(1,2) c G^CE^). (3.1.2) tells us that a)(i,2) = £2(1,2) \ (£2(1,1) un(o,2)). £2(1,2) is defined as a subvariety of G2(E^) by setting the Plucker coordinate x^ = 0. £2(1,1) and £2(0,2) have the additional relations x^ = x^^ = 0 and x00 = x„. = 0 respectively by (2.1.2). Since x_,x„„ = x-_x„, it is not 23 24 J 14 23 13 24 possible to violate the conditions of membership in £2(1,1) and £2(0,2) simultaneously by having x^ ^ 0 4- and x^^ = 0. Thus X eo)(l,2) if and only if x^^ 4 0. We have then 03(1,2) = £2(1,2) n W24 which generalizes to - 48 - Proposition (3,2,2) U(.a1? , , . ,ak) = BCalf. ., ^ n W +k 1 ' ' * k Proof A point in £2^ satisfies certain linear relations according to (2.1.2). Some of these are the defining relations for £2(a^,, . , ,a^) , but the set of extra El'ucker coordinates that vanish on S21 is easily seen to be R. = {x. . ; j . = a.+i and j, < a, + A for X f i} . i J-L* -. . »Jk i i > X Since x e | R. it follows that a k-plane X in W ,.. a.,+1, . . . ,a, +k . x a.,+1, ... ,a,+k 1 k ien 1 k cannot be in any £21, thus 0(a £uCa a ) 0 W lV a1+lf...fak+k r--- k - If X e u> (a, , . ..,a, ) then dim (XnE , .) = i and dim (XnE ,..,)= i - 1, 1 k a.+i a.+x-l I I so we can choose a basis for X such that the i'*1 vector has a.+i1"'1 l coordinate non-zero. Therefore a Stiefel matrix A, for X differs from that in the proof of (2.1.2) only in that each x. .is guaranteed non-zero. X * Si . 'X 1 The k* k submatrix A . is a lower triangular matrix with no zero 3^TIj•••,a^tK entry on the diagonal, thus P (X) det a1+l,...,ak+k = \+l * ° and XeW e d a1+l,...,ak+k' ^ ' " We have already seen in chapter 2, section 2 that £2(1,2) nW.. is an affine space of dimension 3 over the complex .numbers, In particular there is a homeomorphism - 49 - (0(1,2) £ JR6 exhibiting to(1,2) as a 6-cell, This also generalizes to Proposition (3.2.3) 2(a +,.,+ak) u(a^,.,.,ak) is homeomorphic to 3R Proof The Stiefel matrix A of X mentioned in the proof of (3.2.2) has a. k priori £ a.+i entries that are unspecified but A is not a unique representative of X. If we choose A instead so that A ,, = I a^+1,...,ak+k k as usual, or equivalently choose ^ ^a^+1,...,a^+k ^ then we specify an extra -^(k+l) entries corresponding to those on and below the diagonal in A M a . There remain exactly a,+..,+a, entries in a^+1,..., k+k 1 k A' that are free to vary over C. q.e.d. From this it follows easily that ti(a^, . . . ,a^_) is irreducible since it is the closure of oi(a^, . . . ,a^) which is connected. So the Grassmannian G, (E ) is the disjoint union of a finite number of k n J even dimensional cells, and by (3.2.1) the closure of any cell w (a. ,.. . ,a,) is contained in the 2(a,+. . .+a, )^-skeleton. Thus G, (E ) is a finite CW 1 k k n complex, CW The CW chain group (G^(E^), ZZ) is the free abelian group on all the cells w(a^,.,,,a^) such that a^+ ,+ = r. Since all the cells are even dimensional every chain is a cycle and no chain is a boundary, so the homology groups are naturally isomorphic to the chain groups, A homology class depends only on the sequence of integers (a^tltfa^) since any two full flags E, E' of linear subspaces of (i.e. any two bases for E ) are connected by an invertible linear transformation from n •> ~ E^ to itself which induces an invertible, projective linear transformation from JP(AE ) to itself carrying o)(.a , , ,, ,a ) , defined relative to E, n x ic bijectively onto u'(a^ji't>^)» defined relative to E', We have proved The Basis Theorem, Part I The integral homology of G (E ) is freely generated in dimension 2r by JC n the Schubert symbols (a ,.,.,a ) where a +...+a = r. The odd-dimensional X K. X K. groups are all zero. So, for example, the homology groups of G^CE^) that are non-trivial have one generator (0,0), (0,1), (1,2) and (2,2) in dimensions 0,2,6 and 8 respectively and two generators (0,2) and (1,1) in dimension 4. Finding the Belti numbers of G, (E ) is a combinatorial problem. The k. n th 2r Belti number, g is the number of Schubert cycles (a ,...,a ) such that a^+a2+...+a^ = r. If r < k then this is the number b(r) of partitions of r. When r > k the situation is not quite so simple, but we can simplify G E = = as follows, let N = dim ( jc( n)) k(n-k) , we show that g2r $2(N-r) " Let S, be the set of Schubert symbols defined over G, (E ) then the function k,n k n f : S -y S defined by sending the symbol (a1,...,a ) to the symbol iC f TL IC y n X iC (n-k-a^,...,n-k-a^) is a bijection which restricts to a bijection between the set of symbols with sum r and the set of symbols with sum /XT k(n^k) - (an + ,..+a,) = N-r. Thus we have shown that B„ = B„ x. Now " 1 k 2r 2(N-r) T = tne let N = the greatest Integer in then for k = us N-r <_ N' and B2r ^2(N-r) S^Yes those. - 51 - Since G, (E ) is a compact, complex manifold the statement 3n = 3„/lT s k n • ' 2r 2(N-r) follows more generally from Poincare duality. In fact we shall see later that if [ J is theccohomology class of the Schubert symbol and [ ] denotes the dual class then [ ]* - If( )] §3. The Ring Structure in Homology In the case where the figures are linear spaces an enumerative problem can, theoretically, always be solved using the results of the previous two sections. First the solution set is described as a zero-dimensional sub- variety of the Grassmannian, that is, written as an intersection of a finite number of Schubert varieties (intersecting in a finite number of points). Since the description of the solution set usually involves some degeneration into special position these points are counted with multiplicities. We have seen that the intersection number then, counts the number of solutions in the general case since "specializing" can be interpreted as moving within a homology class. The problem is to find the product of two Schubert cycles as a linear combination of other Schubert cycles. The first step is to find the intersection numbers of pairs of cycles in complementary dimensions. To this end let £2 (a.. , . .., a, ) and £2 (b, , . . . ,b, ) be Schubert varieties in G, (E ) so that lk lk k n k k I b. = k(n-k) - I a. . 1=1 1 1=1 1 These varieties do not, in general, intersect transversely, For consider £2(1,2) and £2(0,1) in G2(E4) , We have codim £2(1,2) = 1, codim £2(0,1) = 3, It follows directly from the Schubert conditions 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)), It will be necessary then to find a different representative ft'(b^,,,,,b^) for the class of ft (b^, .,, jb^) so-that the intersection £2(.a^, , , , ,-a^) n ft ' (b^, , . , ,b^) is transverse. We recall the flag E = (E. c E c ... c E ) 0 1 n defined in Chapter 2, section 1, where we chose {e^,...,e^} as a basis for E^, and we define a flag 3E' = (E'cE'c ...cE') 0 1 n where E'. is the linear span of {e ,e }, so that the subscript x n-x+1 n still denotes the dimension. Now define 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^) . If X is in this intersection then it satisfies dxm (XnE , .) > x and a.+x — x dim (XnE' .) > i, for 1 < i < k . D . tl — — — X A more convenient way of writing the second condition is dim (XnE- ) , k-i + 1, k-x+1 Combining the two conditions it follows that (3.3.1) XnE „E» _.+1?M0), isisk x k-x+1 T 53 - since Y= IXnEa ] + [XHE- ] SX i k-i+1 thus: Y has dimension at most k« For C3.3.1) to hold, the definitions of E and ET force a.+i + b. .,.,+k-i+1 > n + 1 1 k-i+1 that is a. +b. . > n-k, l k-i+1 ' arid by virtue of the fact 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 sHa^,...^) n fi'-(b , .. . ,bfc) is empty unless b . = n - k - a. and in t lis case we Ii k-i+1 ^ find the intersection explicitly. We have E E = E E a.+i " bk_.+1+k-i+l a.+i " n+l-(a.+l)' which is simply the linear span of e ... By (3.3.1) it follows that a .+i l j.i»e .1 e ,i ) is a basis for X and consequently dim (XnE ,.) a,+1 a„+z a,+k ^ J a.+i is exactly i. This puts X in coCa.^, , , . ,ak) so X is a smooth point of ft(a^, , , , ,ak) , Letting "{v^, t , , ,v^_} denote the linear span of y^, , , , ,v_., find that - 54 dim (XnE' ,. = dim (XpE' , ) n k +1 1 V ' 'Vi+i = 1 and so X is also a smooth point of 0,' (b , . , , ,b, ) . One shows that the X K. intersection is transverse. By denoting the class 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-ak, n-k-ak_^, n-k-a^) and J = (b^,...,bk). This is the content of The Basis Theorem, Part II (a^,...,ak) = (n-k-ak> n-k-a^^, n-k-a^) * where denotes the Poincare dual. This theorem allows us to find the "coordinates" of an arbitrary 2r-cycle (or its homology class) relative to the basis for the 2rt^ homology group. Let a be any class in H (G (E ), ZZ), then by part one of the basis theorem 6 (a a ) a " I a1,,..,ak l'"" k where the sum runs over all Schubert symbols such that a^+,,P+ak = r. To find 6'- we intersect both sides with (n-k-a , , , , ,n-k^a1) whose intersection - 55 - number is one with (a^?,,,,a^) and zero with any other class in the sum. That is 7a,(n-k-a ,,,«,n-k-a )\ = § The integers S were referred to by Schubert as the degrees of V in a1,,..,ak the case that a was the class of an irreducible subvariety V of the Grassmannian. Suppose W is another irreducible subvariety of complementary dimension and let y, , be its degrees, that is "•^ > • • •» [WJ b (bl ,bk) •^bi>---» k '"" where [W] denotes the homology class of W and the sum ranges over sequences such that b,+. . .+b, = N-r. Then we have 1 k cL • • • cl 1''*"' k n-k-a^,...,n-k-a^ where again the sum ranges over all Schubert symbols (a^,...,ak) in dimension r. In the case where k=l i.e. G, (E ) = Pn ^ this equation reduces to k n Bezout's theorem. The foundations of the Schubert calculus are set down with the basis theorem together with the following two theorems which allow us to compute the product of two arbitrary Schubert cycles in terms of a particular set called the special Schubert cycles. Let - 56 - a'(d) = n(n-k-1,,,,,n-k-l,n-k,,,,,n-k) where the number of (n-k^l)'e that appear is exactly d, and let c^ denote til the homology class of a(d), a(d) is called the d special Schubert variety and we can now state Pieri's Formula (a^ .. .,ak) • od = £(1^, .. . ,bfc) where the sum ranges over all Schubert symbols (b1, ... ,b ) satisfying a. .. < b. < a. for 1 < i < k and satisfying l-l —1—1 — — J to codim - (b.. b, ) = codim„ (a..,...,a.) + codim., a (Ll K. (Ll; lc To illustrate this we compute the self intersection of (1,2) eH6(G2(E4), 7L). By definition (1,2) = a2, and codim (1,2) = coding = 1. (1,1) and (0,2) are the only Scbubert symbols with the desired codimension and they both satisfy the inequalities required by Pieri's formula. Hence (3.3.2) (1,2) (1,2) = (1,2) ai = (1,1) + (0,2). There is a companion to Pieri's formula which shows that HA(Gk(En), 2Z) can be generated as a ring by the classes of special Schubert cycles, with the intersection cycle as the product, It is - 57 ~ a a a 1 a Ca. a 1' "'*' k^ a 2 a,+k-2 a, k k where a. is defined to be zero if i < 0 or i > k. l Both of these theorems are proved complex analytically in Griffiths and Harris. Their: treatment would be difficult to improve upon so we refrain from reproducing the proofs here. Let us apply our new found techniques to the simple enumerative problem of finding the number of lines simultaneously meeting four given lines in 3 general position in TP . We have already seen that the solutions are the points in the subvariety 4 v = n ^[L,,JP3] i=i 3 of G^(TP ) = G^iE^), where L^,L2,L^ and L^ are the four lines. Now 3 ii[L^,TP ] is the same variety as £2(1,2), by the remark following (2.1.2), where L_^ = TP(E ^) and TP5 = TP (E^) . Consequently i, the number of points in V counted with their multiplicities, is the intersection number of the fourfold self-intersection of (1,2), that is i = (1,2)4«4C1,2)2,C1,2)2)' = -<(1,1) + CP,2),C1,1) + C0,2)}, The last expression follows from (3,3.2) and can be computed as follows; - 58 - {(1,1) + (0,2),(1,1) + (0,2)} = <(1,1),(1,1)> + l{ (1,1), (0,2),} + { (0,2), (0,2)} and the second part of the basis theorem applies killing the second term, while the first and last terms are both 1 since (1,1) and (0,2) are Poincare self-dual. In agreement with our previous solutions of this problem we find that there are two lines in general intersecting four given lines in general position in complex projective 3-space. Let us compute a higher dimensional example. We wish to find the number of 2-planes in projective 5-space that intersect 9 given 2-planes. The condition that a 2-plane X meet a given 2-plane f[ is dim (X n H2) > 0 but since dim (X n IP5) =2, then dim (X n TI-j) S 0 and dim (X n TJ.^) > 1 by a property of intersection sequences, where ™ 5 n2 - n3 - n4 - ' and dim n_ = i. Thus X lies in the Schubert variety 5 fl[n2,n4, TP ] = fiflP (E3),TP (E5),TP (E6)] = £2(2,3,3) which is the first special Schubert variety on G^CE^) = G2(JP"'), The number we wish to compute is equal to the nine-fold self intersection number of the special Schubert cycle = (2,3,3) , Applying Pieri's formula,, the square of is (1,3,3) + (2,2,3) = o2 + c2 where a2 denotes the cycle (1,3,3), The third power of is CT + a h ' i 2 • °i which by Pieri's formula is (0,3,3) + 2(1,2,3) + (2,2,2). Recursively we find oj = 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 part II of the basis theorem we can evaluate + 2x6 + °1 = °1 ' °1 = 3X5 3x5 = 42. As mentioned in the introduction, it has been shown, in general, that the number of k-planes in n-space meeting h = (k+1)(n-k) general (n-k-1)-planes is h!k!(k-1)!.•.3!2! , n!(n-1)!...(n-k)! and our two examples are special cases of this, - 60 - Chapter IV MORE RECENT DEVELOPMENTS §1 The Hasse Diagram An object that contains a surprising amount of information about the Grassmannian is a certain lattice called the Hasse diagram. It can be defined as the lattice associated to the set of Schubert varieties partially ordered by inclusion. More precisely let (a^,..,a^) be a Schubert sub- variety of G. (E ) then the Schubert symbol (a1,.,,,a.) defines an integer K. XI X K. point in ]R . Define (a1,...,ak) A (b1,...,bk) = (min [a^b ], min [a^.b ]) , (a^, .. . .a^. v (b1>...,bk) = (max [a^b.^], min [a^jb^]). It is easy to check that A and v form the greatest lower bound and least upper bound respectively, making H, = {(a..,...,a );ft(a ,...,a1 ) is a Schubert subvariety of G (E )} K,n x K X K K n into a distributive lattice. When dealing withi.the first model of H, , k,n A and v have geometric meaning in that fi(a1}...,ak) A fl(b , ...,bk) = fi(a ,...,ak) n S2(b1> . . . ,bfc) id(a ,. .. ,a, ) v H(b , ..,,b,) = smallest Schubert variety containing J. K. X K. J2(.a1? , ,, ,ak) u fi(b1? , , «,bfc) , Before exploring the structure and other models of H we draw the tC y n diagrams in a few low dimensional cases, Many appealing patterns present themselves straight away. We deal with some that have geometric significance. The homology basis can be picked out - 62 - (2, £,3) 3.fc C 0,0,0) - 63 - easily, The generators for R^r (G^ (E^) , ZZ ) are the points of ^ lying on k the hyperplane x^+x^+ + x^ = r in 3R , In our examples this hyperplane is a horizontal line, so for 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 with generators (1,2,2), (1,1,3) and (0,2,3), etcetera. Thus the Betti numbers can also be read easily in the same way, so for example the even Betti numbers (B^,^, • • • >®ig) of G3^E6^ are -*->l>2>3,3,3,3,2,l,l respectively. The odd ones are, of course, all zero. In all the examples drawn, we see that the "top half" and the "bottom half" of the Hasse diagram have the same shape. This is true in general and k is a consequence of Poincare duality. The hyperplane II in 1 k n k V v - < - > L i " 2 i=l intersects H, exactly when dini (G, (E )) is even. In this case II n H k,n J C k n k,n is the set of Schubert symbols that are Poincare self-dual. A point above the hyperplane is related to its Poincare dual below and this pair of points defines a line segment whose mid-point is in II. We do not, however, see any symmetry happening from left to right in general. In the case H and H and in the case of H s and H , the diagrams are the reflections of each other in a vertical axis. This reflection is the Hasse diagram's interpretation of the canonical isomorphism G, (E ) •== G , (E ) . We mention a technical peculiaritJy here; even though k n n-k n G, (E ) and G , (E ) are canonically isomorphi1 c their corresponding Hasse k n n-k n diagrams are not necessarily the same, yet G , (E ) and G ,(£**), which are 0 J ' J n-k ' n n-k n ? not canonically isomorphic, have precisely the same Hasse diagram, Thus H depends only on the integers k and n, and H = H if and only if K. y Tl iC y Ti T1™*JC y Ti - 64 - k = n-k, witness H^ ^ and ^. The Hasse diagram then distinguishes between the Grassmannian and its' dual :but doesn't care /where the vector space comes from; in particular the Hasse diagram will be the same over any ground field IF . The diagrams have been drawn so as to show how H and H fit into H^ ^ and how H^ ,. and H^ fit into H^ ^, More generally the Hasse diagrams of G. (E ) and G . (E ) fit into H „ where m = max (k,n-k). This fit is a k n n-k n m,2m ' useful device to help understand the Hasse diagram's interpretation of duality explicitly. The same argument used in the latter part of example (2.1.4) shows that fi(0, . .. ,0,a.., . .. ,a, ) of G. , , (E ,,) is isomorphic to 1 k k+d n+d f!(a, a.) of G, (E ) where d is the number of zeroes .in the first Schubert 1 k K. n symbol, This induces an injection of sets H f r 311 1 k d k,n ^ \+d,n+d ° ^ ^ > via (4.1.1) (al5...,ak) (0,.. .,0,a ,...,a ) which preserves both bounds. If we assume k < n-k then H , embeds in a n-k,n different way into a higher dimensional diagram. In fact H , —>• H . , for all 1 < k < n, 1 < d' n-k,n n-k,n+d ' via (a^, ,, ,, a^)•—>• (a^,,,,, a^) which is induced from considering an (n-k)^plane Y in E cE ,, to be an * n n+d (n-k)-plane in -E If we now choose d to be the integer such that ^k+d n+d = ^n-k n+d 1,e* d = n*"2k we find that the Hasse diagrams of the - 65 - Grassmannian and its dual are distinct sublattices of H , 0, lN, n-k,2(n-k) The diagrams H\ ^ are precisely those which do display the "sideways" symmetry and we call them the self-reflexive Hasse diagrams, Our desire is to find an explicit map A : Hn-k,2(n-k) "* Hn-k,2(n-k) which is a lattice isomorphism, by which we mean a bijection of sets preserving both bounds. We wish it to have the property that the image of H K , n is H n—lc, n, and that it have order two. Such a map we call a reflection, and if CT eH „, , then we call the image the reflection of o. To n-k,2(n-k) c accomplish this let r = n-k and N = 2(n-k) = 2r, choose a flag T£ = (EQ C E2 c • • •c anc* from it define a flag 3D in E^ via D. = E° . i N-i where V° denotes the annihilator of V. Let X be a point in ai(a, ,...,a ) 1 r N o nD and consider its intersection sequence {a^}^ and let 3^ = dim (X j)- We have a± = dim (X n E ) thus 0 N-a± = dim ([Xn E^ ) = 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 is SN-i ^ ai+ ^"^ or, on replacing i by N-i (.4.1.2) 0. = aM . - (r-i) . Thus the intersection sequence of X° with respect to JD can be computed from the intersection sequence of X with respect to E . Denote the lag sequence of {g.}1? _ by (b..,...,b ) and define l 1=0 1 r A(ai,..;,ar) = (b ,...,b ) • The fact that the reflection has order two is clear from the construction and thus the fact that it is a bijection of sets. That the bounds are preserved can be proved by taking a close look at the intersection sequences and following indices. The details provide little in the way of geometric intuition and are omitted here. G, (E ) embeds in G (E ) as the Schubert variety ft(0,...,0,n-k,...,n-k) where the number of zeroes is n-2k. We compute its reflection. 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) V t \ I, > " • -y—• ' t i- (n-2k) terms (n-k) terms k terms with respect to IE and according to (4,1,2) - 67 - with respect to B , Thus X°ew(k,k,,,,,k). Thus we have shown that the reflection of (0,,.,,0,n-k,,,,,n-k), which represents G (E ), is (k, , ,,,k) which represents G , (E" ), It follows that the reflection map restricts to n-k n an isomorphism between H, and H , k,n n-k,n The reflection map reconciles the two choices in the literature of the basis for the cohomology ring (or, equivalently the homology considered as a ring with the intersection product). Our choice is that of Griffiths, whereas others, notably Kleiman and Laksov, choose the special Schubert cycles to be of the form (j,n-k,n-k,.,.,n-k) for 1 < j < n-k. So, where we have k special Schubert cycles on (G, (E ) the other choice K. n counts (n-k) special Schubert cycles. Again by looking at the intersection sequences it is easy to see that these (n-k) alternate Schubert cycles are simply the reflections of our choice of (n-k) special Schubert cycles on G . fe ) . n-k n One would hope that Giambelli's formaula is compatible with the reflection. Since calculations in higher dimension become very cumbersome very quickly we show the truth of this only in the example of G^iE^). Our choice of special Schubert cycles then is = (1,2), = (1,1), whereas the other choice is a-^ = (.1,2), = (0,2). Now aA CTA-1 (,2-ui2-A) = (y,A)* = * aA ' au Vl ' Vl aA ' % y+l . y since at least one of A - 1 and y+l is outside the range stipulated by Giambelli's formula. Using the reflections instead we have - 68 Note that codim a. codim o. for i 1?2 and so by Fieri's formula. x 1 o • a a • A = a '•- a a = a X • a A A ..y y A y In general Pieri's formula is preserved under reflection since the only property of a special Schubert cycle it uses is its codimension, which is preserved, Returning to the Hasse diagram we call a Schubert symbol a and immediate predecessor of 3 in case a < 3 and there does not exist y such that a < y < & where < is the total ordering on the Hasse diagram. Geometrically an immediate predecessor of a Schubert variety ft is a Schubert subvariety of codimension 1 in ft. Proposition (3.2.1) can be restated then as uj(.a , . . . , a, ) = ft (a , ...,a ) \ all immediate predecessors of ft (a , . . . ,a, ) . Since a)(a^,...,a^) is smooth, the singular locus of ft(a^,...,ak) is contained in the union of its predecessors. We describe precisely which ones, but first let us look at the example of ft(l,2). In H . the symbol (1,2) is the only one with more than one immediate predecessor, moreover, ft(1,2) is the only singular Schubert variety in•-G (E^). This is not a coincidence. In 1974, Svanes published a paper entitled Coherent Cohomology on Schubert Subschemes of Flag Schemes and Applications in which he constructed an explicit resolution of the singular locus of an arbitrary Schubert variety over an arbitrary ground field, The complete proof of this is highly technical and beyond the scope of this discussion, we only quote the result, - 69 - Theorem (.4.1.3) Let kQ be zero and a^ > 1, If k^. is defined inductively by ak. . < \. • +1 " ak. ,+2 = = ak. < \.+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 is simply a convenience since To illustrate this we consider an example large enough to see what is 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) is also a special case of (4.1.3) i.e. Sing £2(1,2) = £2(0,0) . A consequence of this can be seen on the Hasse diagram, namely Corollary (4.1.4) A Schubert variety is singular if and only if the corresponding symbol in the Hasse diagram has at least two immediate predecessors, - 70 - Proof: Suppose (a,, ,n(a,l has at least two immediate predecessors then there exist 1 < 1 < j < k such that a. T < a. =a. . < a., i-l i J-l J Consequently ft Ca, a. . ,a .-1, . , . ,a . ,-l,a. -l,a.,„,.,, ,a,) 1 5.1-1' i j-l j-l J+2 k is contained in the singular locus of ti(a^, , ,, ,a^) and is non-empty. Conversely suppose (a..,..,,a,) has only one predecessor. Let a. be the X JC X first non-zero integer in (a^,...,a^) then the unique immediate predecessor is CO,...,0,a -l,a ,...,ak). We claim that a. = a. for all i > i, for if not let a. be the first integer i J i 5 in the Schubert symbol (0, . . . ,0,a .,...,a, ) such that a. > a. then J k i J (.0, . . . ,0,a., . . .,a ,a -1, .. .,ak) would be a different immediate predecessor. So fi(a^,...,a^) is of the form ft(0, . . . ,0,a,. . . ,a) which,bby example (2.1.4), is the Grassmannian G..(E ,,) and is therefore smooth, q.e.d. In the course of the proof we have established Corollary (4.1.5) A Schubert variety is smooth if and only if it is a Grassmannian. It is now easy to count the number of smooth Schubert varieties in G,(E ). There are Q, (0, . . . ,0) and each ft(0, . .. ,0,a, . . . ,a) where a occupies the last i places for all 1 < i < k and runs from 1 to (n-k). This makes k(n-k)+l smooth Schubert varieties, This also gives us the curious fact that the number of singular Schubert varieties is exactly the codimension of Gk(En) as a projective variety via the Plucker embedding, At the time of writing we see no intrinsic geometric reason for this though we believe there is one, - 71 - The Plucker coordinates themselves, however, form another model of the Hasse diagram by labeling the vertex (a , ,r,,a.) as JX , and this 1 K ^l ' ' ' ' ? 3]j does have geometric content by virtue of Proposition (3,2,2) i,e, a point 0 of w(a1,,,,,ak) is a point of n (a , ,'f , ,afc) where x^ +1 a +k ^ ' 1 ' ' ' '' k It is possible to show that, for a self-reflexive Hasse diagram H , Poincare duality commutes with reflection, i.e. the symbol A a * (a^,.,.» k) is unambiguous. Suppose 1< X <... < A, ^ 2k and suppose I & 1 < y^ < ... < < 2k is the complementary sequence, by which we mean that (y^,...,yk) is the set {l,2,...,n} \ {A^,....X^} arranged in ascending order. Now suppose that (a ,...,a ) is the vertex of H- corresponding to X iC R) ZK x and that (bb,) corresponds to x . We conjecture that A^. ••..,Ak 1 K- y^,...,yk A * (a1,...,ak) = (b1,...,b ) . If I and J are complementary sequences then the quadratic relation k + P q XX = V (-1) x s X <" p=l 1 p ^,q k 1 q Ap k does not collapse on W^. n W for 1 < q < k, thus the conjecture might help to find independent quadratic relations locally, of which there should be (k) - k(n-k) - 1 = the number of singular Schubert varieties. - 72 - §2 Concluding Remarks The intersection theory used here relied heavily on the fact.that we were working over the complex numbers. It was developed by Lefschetz and applied to the foundations of the Schubert calculus by van der Waerden in 1930. Ehresmann found the cell decomposition in 1934 and developed some general results about cell complexes, that are now standard, to prove the basis theorem. All of this however was topological. Hodge produced the first purely algebraic intersection theory in 1941 and 1942 with the papers The base for algebraic varieties of a given dimension en a grassmannian variety and The-intersection formulae for a grassmannian variety. He proved the basis theorem, and Pieri's formula for an arbitrary special Schubert cycle o^, while van der Waerden had first shown the case where 1=1. Hodge then used Pieri's formula to prove Giambelli's formula. A great many intersection theories were developed after Hodge's and in the following we make no pretensions of completeness. Perhaps the most notable was the Chow ring, defined as follows: let C (V) be the free abelian group on all irreducible subvarieties of V, a projective variety over an algebraically closed field. If X and Y are irreducible subvarieties of V then they are said to intersect properly if codcodim:(Z) = codim (X) + codim (Y) for every irreducible component Z of X n Y. The abstract definition of intersection multiplicity that was mentioned in the introduction was invented by.Se.rre in 1965 in a paper entitled Algebfe locale - multiplicites, and is defined in terms of homological algebra. We denote the multiplicity of a component Z of X n Y by I(X,Y;Z) and so it is possible to define a product - 73 - X . Y = I I(X,Y;Z)Z , ZcXnY whenever X and Y intersect properly. One now considers various equivalences on C (V) that guarantee choices of representatives for each pair of equivalence classes, so that the representatives meet properly. There is a hierarchy of these equivalences and the strongest one.is called linear or rational equivalence. Two varieties are linearly equivalent if they are both members of an algebraic system of subvarieties parametrized by TP \ The Moving Lemma For any two varieties X,Y-on V there is a variety Y', linearly equivalent to Y, such that X and Y' intersect properly. If X,Y,Z are varieties such that X is linearly equivalent to Y then whenever X • Z and Y • Z are defined A X . Z is linearly equivalent to Y • Z. The quotient of C (X) by linear. equivalence is the largest ring for which the intersection product is defined everywhere. This ring is called the Chow ring. Linear equivalence can be weakened. Instead of having the continuous family parametrized by we have it parametrized by a quasi-projective variety U. The resulting relation is called algebraic equivalence. The hierarchy is as follows: linear equivalence=>algebraic equivalence=>homological equivalence. Here homological equivalence means membership in the same Weil cohomology class, where a Weil cohomology is an invariant on varieties over a characteristic zero field that behaves formally like singular cohomology on manifolds. There is even weaker equivalence define'.d as follows; consider - 74 - C*(V) + Cn(.V) + 2Z n ' * where C (V) is the subgroup of C (V) generated by subvarieties.of'codimension * n (i.e. points)v, the first map is the projection (C (V) is a graded group), and the second map adds up the coefficients of the linear combinations: The second map is called the augmentation.' If X • Y is defined, then its image under the composite map is an integer called the intersection number of X and Y. It ignores components of X n Y of positive dimensions counting only points with their multiplicities. If homological equivalence =>numerical equivalence and indeed it does, but the opposite implication is more interesting. Grothendieck's 1958 publication Sur quelques proprietes fondamentale en theor'jefetes intersections showed that there is a certain general class of varieties, which contains Grassmanmians and flag varieties, with the property that numerical equivalence =>homological equivalence. He actually proved a more general result but it is not needed here. Laksov, in 1972, constructed an intersection theory over an arbitrary ground field in his paper entitled Algebraic Cycles on Grassmannian Varieties. He proved, using this theory, the basis theorem and versions of both Pieri's - 75 - and Giambelli's formulae. In contrast with Hodge's-method, Leksov proved Giambelli's formula first and used it to prove Pieri's formula. Using Lefschetz' intersection theory we solved two examples in enumerative geometry where the figures were linear spaces. Kleiman has shown that this can always be done in characteristic zero, and more generally for any figures where the general linear group acts transitively on the parameter variety. This is not the case for conies; the general linear group in this case has four orbits, namely the set of non-singular conies, the set of pairs of distinct lines, the set of double lines with distinct foci and the set of double lines with double foci. In certain cases however it is possible to solve an enumerative problem if the figures are not "straight." For example, Kleiman and Laksov prove that the number (originally found by Schubert) of 4 lines common to two quadric hypersurfaces in ]P is 16. Also, in their 3 article Schubert Calculus, they show that the number of lines in IP which simultaneously intersect four given curves C^, C^, C^, C^, if finite, is equal to 261626264, where 6^ is the degree of C^ and the number is counted with multiplicities. The program given by Hilbert's fifteenth problem is immense, there is much classical work still to verify. Some aspects of the problem have been solved repeatedly, but, in truth, we must still consider Hilbert's fifteenth problem unsolved. - 76 - BIBLIOGRAPHY 1. Borel, A., Linear Algebraic Groups, W.A. Benjamin Inc., 1969. 2. Chern, S.S., Complex Manifolds Without Potential Theory, D. Van Nostrand Co. Inc., 1967. 3. Dold, A., Lectures on Algebraic Topology, Springer-Verlag, 1972. 4. 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