Schubert calculus: an algebraic introduction

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A Sheila e Giuseppe

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Preface The goal of these notes is to let the Reader get familiar with a natural algebraic formalism equivalent to Schubert Calculus on Complex Grassmannians, basing on the following Theorem. The Chow intersection (or integral cohomology) ring of the complex grassmannian variety G(k, n), parametrizing k-dimensional vector subspaces of Cn , can be realized as a commutative ring of endomorphisms on the k th exterior power of a free Z-module of rank n (Cf. [23]). The explanation of the above result, as well as a list of consequences, which seem relevant at least from a “pedagogical” point of view, will be postponed to the Introduction. However, it is worth to anticipate that, from a purely algebraic point of view, Schubert calculus for grassmannians G(k, n), for all 0 ≤ k ≤ n and all n ≥ 0 at once, amounts to study the formal properties of a suitable algebra homomorphism ^ ^ Dt : M −→ M [[t]] (1) V from the exterior algebra M of a free Z-module M , of infinite countable rank, to the algebra of formal power series in one indeterminate t over it. In a sense, Schubert calculus can be proclaimed with the slogan: ^ Dt (α ∧ β) = Dt α ∧ Dt β, ∀α, β ∈ M, which is the explicit way to phrase that the map (1) is an algebra homomorphism. These notes have been written as a ready-for-use reference for the minicourse Schubert Calculus: an Algebraic Introduction given i

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ii at IMPA during the 25◦ Col´ oquio Brasileiro de Matem´ atica. By no means they can be intended as substitutes of the many good textbooks on either the geometry or the intersection theory of complex grassmannian varieties. For these subjects there is a plenty of canonical references, rich of geometrical insight, starting from the beautiful [39] up to [27] and [16], or [22]. For a hot update of the state of arts, the Reader should also consult the very recent volume edited by P. Pragacz (including his paper [65] therein). These notes are just a very personal (and, therefore, not necessarily the best) point of view which the author look at those subjects with: nothing more than a pair of coloured glasses to read the references quoted in the bibliography, whom the Reader is really referred. Although the intersection theory of grassmannian varieties is a fairly delicate subject, certainly requiring some mathematical maturity, this formulation shows that one can learn Schubert calculus with minimal pre-requisites (basic linear algebra and calculus) without even knowing what is it. In particular, any reader, even the less experienced one, can read Chapter 5, the core of the notes, independently from the rest of the exposition. Acknowledgements.1 My first debt of gratitude is with my distinguished friend Israel Vainsencher, who last year suggested me to submit a proposal for a minicourse on this subject, to be given during the 25◦ Col´ oquio Brasileiro de Matem´ atica, at IMPA. I could profit of his corrections and remarks because he was the only person I could send my very prelyminary informals drafts without feeling (too) ashamed. The friendly support of Israel was strengthened by the convinced one of Abramo Hefez, whom I want to adress a feeling of gratitude, too. My original proposal to Comiss˜ ao Organizadora do Col´ oquio was somewhat unbalanced and I could improve it thanks to the friendly suggestions and remarks by Eduardo Esteves, who 1 This work has been partially sponsored by INDAM-GNSAGA, by the Italian Ministry of Education, National Project “Geometria sulle Variet` a Algebriche” (Coordinatore Prof. Sandro Verra), and by Dipartimento di Matematica del Politecnico di Torino. The author is especially grateful to the Comiss˜ ao Organizadora do 25◦ Col´ oquio Brasileiro de Matem´ atica, to the Instituto de Matem´ atica Pura e Aplicada and to brasilian CNPq for its generous financial support; he is also indebted with Priscilla (DDIC do IMPA) for her patient assistance during the preparation of the manuscript.

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iii influenced the organization of the contents, whom I am grateful too. I am also strongly indebted with Aron Simis (who gave me valuable hints) and Francesco Russo that, together with Israel and Eduardo, have made possible my visit to Brazil. The subject of these notes, based on [23], has been widely discussed with Dan Laksov whose wise advises, sharp criticism and friendly support are priceless. Also I want to thank Gary Kennedy for discussions on the quantum cohomology side, and Giorgio Ottaviani for discussions on the representation theory one, as well as Piotr Pragacz, who suggested me new difficult but exciting directions to push forward my current research interests. I got a precious help from my collaborator Ta´ıse Santiago Costa Oliveira (obrigado!), Ph.D. student at Politecnico di Torino, and Caterina Cumino. Not only the latter shared (as still does) an office with me, but also many delightful mathematical conversations, including the subject of these notes, which helped me a lot to improve the exposition. Last, but not the least, I warmly thank all the readers, without blaming the others, who will forgive the several misprints, errors and/or omissions, certainly left, here and there, along the text. These two

hidden

notes have co-

authors:

my wife

Sheila and my son Giuseppe. The former solved all the practical problems I should have dealt myself in the daily life, letting me spending, with lovely comprehension, almost all my home time playing with Mathematics.

The latter put abundant

grains of joy with his wide smile, brighting even during cloudy days. Without them these notes could have not been written and for this reason, and many others, to them are dedicated. ♥

Sangano, 30 de Abril, 2005

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Contents 1 Propaganda 1.1 Revisiting (some) Calculus . . . . 1.2 Wronskians and their Derivatives 1.3 Algebraic Schubert Calculus . . . 1.4 Grassmannians . . . . . . . . . .

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3 3 7 11 15

2 Preliminaries 20 2.1 Combinatorics . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Exterior Algebra of a Free Module . . . . . . . . . . . 25 2.3 Review of Intersection Theory . . . . . . . . . . . . . . 28 3 Frames and Grassmannians 38 3.1 Warming Up . . . . . . . . . . . . . . . . . . . . . . . 38 4 Frames and Grassmannians 45 4.1 Complex Grasmannian Varieties . . . . . . . . . . . . 45 4.2 Schubert Varieties . . . . . . . . . . . . . . . . . . . . 50 4.3 Intersection Theory on Gk (V ) . . . . . . . . . . . . . . 55 5 Schubert’s Algebra 5.1 Hasse-Schmidt Derivations . . . . 5.2 Shift Endomorphisms . . . . . . 5.3 Schubert Derivations . . . . . . . 5.4 Pieri’s Formula for S-Derivations 5.5 Giambelli’s Problem . V. . . . . . k 5.6 A Presentation for A∗ ( M , D)

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62 63 66 70 74 76 81

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CONTENTS 5.7 5.8

The Finite Case . . . . . . . . . . . . . . . . . . . . . . Giambelli’s Formula . . . . . . . . . . . . . . . . . . .

6 Miscellanea 6.1 The Intersection Ring of G(2, 4) . 6.2 Playing with S-Derivations . . . 6.3 Degree of Grassmannians . . . . 6.4 Wronskians Correspondences . . 6.5 Small Quantum Cohomology . .

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84 91

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Introduction Schubert Calculus for grassmannians 1 has an ancient and venerable history. It is related with the deep investigations in enumerative geometry pursued by Schubert2 in his celebrated treatise [68], which dates back to 1879. The first systematic study of what, in modern language, would be called intersection theory on the grassmannian variety of lines in the 3-dimensional projective space, is due to him. Schubert’s work was somewhat revolutionary for his time and provided new powerful techniques to deal with many enumerative questions. The most popular is perhaps how many lines do meet 4 others in general position in the projective 3-space? Schubert’s methods, based on heuristics, although ingenious, degeneration arguments, cried however for more mathematical rigour, explicitly claimed by Hilbert, who so entitled his 15th Problem: Rigorous Foundation of Schubert’s Enumerative Calculus. In Hilbert’s words (borrowing the english translation from [38], p. 327), the problem consisted in establishing “. . . rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert ([68]) 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.” 1 2

After Hermann G¨ unther Grassmann (1809-1877). Hermann C¨ asar Hannibal Schubert (1848–1911).

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Schubert’s work was continued and generalized, at the beginning of last century, by two italian mathematicians, Pieri and Giambelli ([28], [29], [30] and [58]), whose formulas allow in principle, speaking a modern language, to explicitly compute the interesection product of two arbitrary generators, said to be Schubert cycles, of the Chow ring of a grassmannian (see Section 4.3). They were far, however, from giving answers to Hilbert’s problem, which had to wait for the modern developments of intersection theory and/or of the singular cohomology of complex projective algebraic varieties. Readers interested to follow this nice historical path, should not hesitate to look at the surveys [38] and [39]. Nowadays, talking about Schubert Calculus, without any further adjective, is too vague to indicate any specific mathematical subject. Schubert calculus for grassmannians has been generalized to flag varieties (see [18] and [8]) and, more generally, to homogeneous spaces (quotients of an algebraic group modulo a parabolic subgroup; see [6] or [76] for general vocabulary). There are also versions of Schubert calculus for other types of grassmannians (orthogonal, symplectic, see [62], [66]) of vector spaces equipped with an additional structure (metric, symplectic, etc.), without talking about the rich flourishing of the literature regarding combinatorial aspects of small quantum Schubert calculus initiated with Bertram in [4] and continued in a series of papers (e.g. [3], [5], [9], and references therein). More than that, the structural constants of the cohomology ring of the grassmannians appear in the representation theory of the general linear group Gln (C) (see [20], [75], and the beautiful paper [77]), as well as in the more classical and fascinating theory of the symmetric functions (see [52] or the more recent [53]), where the k × k Vandermonde determinant plays the same role as the fundamental class of a grassmannian G(k, n). People interested to the marriage of grassmannians with physics should have a look at [26] and, of course, to [81]. Regarding more specifically these notes, they aim to be a walk, back and forth, between the land of Schubert’s algebra (governing the intersection theory of the grassmannians) and that of Grassmann’s algebra (the exterior algebra of a module) through the bridge of wronskians3 . Metaphores aside, the goal is to introduce the Reader to an 3 After

the polish mathematician J. M. Hoene-Wro´ nski (1778-1853).

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abstract algebraic model equivalent to Schubert calculus for grassmannian varieties, whose corner stones, Pieri’s and Giambelli’s formulas, can be rephrased in terms of derivatives of wronskians determinants of a system of linearly independent (C ∞ or holomorphic) functions of one (real or complex) variable. It is easily checked that the ith -derivative (i ≥ 1) of a wronskian turns out to be an integral linear combination of certain determinantal expressions, said to be, imitating a terminology used in [59], generalized wronskians (Cf. Section 1.2.1). Pieri’s formula then corresponds to differentiate wronskians, while Giambelli’s formula amounts to integrate the generalized ones, as explained with more details in Chapter 1, aimed to advertise the contents of Chapter 5, the core of the notes. The algebraic formalism there developed turns out to be particularly flexible, due to the natural identification between the Chow (or homology) group A∗ (Gk (V )) of the grassmannian of k-planes of V , Vk and the k th exterior power Mn of a free Z-module Mn of rank n = dim(V ). Schubert calculus is then described through a family V D = (D1 , D2 , . . .) of derivations of the Grassmann algebra Mn V that, when restricted to k Mn , generate a commutative subalgebra Vk Vk A∗ ( Mn , D) of EndZ ( Mn ), which is isomorphic to the intersection ring A∗ (Gk (V )) of the grassmannian of k-planes of a n dimensional vector space. Moreover there is the isomorphism k k ^ ^ A∗ ( Mn , D) · ǫ1 ∧ . . . ∧ ǫk ∼ Mn , =

(2)

translating Poincaré duality for grassmannians (Cf. Section 4.3.2). Within this framework, Pieri’s formula is nothing else than a consequence of Leibniz’s rule enjoyed by the operators Di ’s (which correspond to the special Schubert cycles σi ), while Giambelli’s formula, which formula (2) is based on, turns out to be a consequence of the following integration by parts (Lemma 5.5.3) Di α ∧ ǫj = Di (α ∧ ǫj ) − Di−1 (α ∧ ǫj+1 ), V holding for each α ∈ Mn .

Rather than working with the k th exterior power of a finitely generated free Z-module Mn , in Chapter 5 one works directly on the

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V exterior algebra M of a free module M of infinite countable rank. This corresponds to deal with all the grassmannians at once, rather than a single one, making things easier and more elegant. Everything V is described in terms of a particular Hasse-Schmidt Derivation on M (said to be Schubert for obvious reasons; see Sections 5.1 and 5.3), permitting to summarize Schubert Calculus formalism (and hence Pieri’s and Giambelli’s formulas) via the holding-by-definition equality: ^ Dt (α ∧ β) = Dt α ∧ Dt β, ∀α, β ∈ M. One should dutifully remark that the picture of Chapter 5 has been rephrased, and considerably refined, by Laksov and Thorup ([48]). Using suitable symmetrizing operators (analogous to taking residues of formal Laurent series) and the notion of universal splitting algebra, they are able to give very transparent proofs of the analogous of Giambelli’s formula, discussed in Section 5.8, even for grassmannian bundles. More than that, those authors are able to show ([49]) how it is possible, using the theory of splitting algebras, to prove that the picture exposed in Section 5 is indeed the intersection theory of the grassmannian variety without relying on any previous knowledge of Schubert Calculus! These book-shaped notes are organized as follows. Chapter 1 is aimed to convince the Reader that Schubert Calculus for grassmnannians can be phrased in terms of everybody’s first mathematical experiences and that, to learn it, is not necessary to know what is it. Most of the examples of Chapters 6 could be in fact computed by first year’s undergraduate students. Chapter 2 should be skipped by most readers, as it is a list of quickly collected basic definitions and preliminary results (without proof), to be used mainly as a vocabulary. Chapter 4 is about an introduction to the notion of the grassmannian as a holomorphic variety. This also may be avoided byt experienced readers. However the presentation aims to prepare the way to the, although easy, abstract formalism developedVin Chapter 5. What in k the latter is a basis element ǫi1 ∧ . . . ∧ ǫik of M , in Chapter 4 is a Pl¨ ucker coordinate or a section of the top exterior power of the dual of the tautological bundle over the grassmannian. Finally, Chapter 6 is a miscellanea of fragmented examples, intended to show that the for-

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malism offers indeed some computational advantages. Section (6.1) fully works out the computation of the intersection theory of the grassmannian of lines G1 (P3 ), while Section 6.5 is devoted to algebraically analyze what in the literature is known as quantum Schubert calculus of G(k, n). It turns out that the latter can be seen as Schubert calculus in a grassmannian G(k, N ), with N sufficiently large. The examples about degree of grassmannians, shown in Section 6.3, cry for the search of a more general pattern. The degrees of Schubert varieties in the Pl¨ ucker embedding of the grassmannians are very well known (see e.g. [36]). However, the Hasse-Schmidt presentation suggests a search for degrees through recursive formulas, as illustrated by the shown experiments. Using the algebraic yoga introduced in this book, one may easily show that the degree of the Pl¨ ucker image (Section 4.1.5) of the grassmnannian variety G(3, n) is 3(n − 3) 3(n − 3) 3(n − 3) d3,n = d2,n−1 + − d2,n , n−3 n−5 n−4 having denoted by dk,n the degree of G(k, n). A similar, although slightly more complicated formula expressing d4,n as a suitable linear combination of d3,n , d3,n−1 , d2,n and d2,n−1 , is displayed in Example 6.3.4. Computations get trickier for d5,n and one wonders if it is possible to find a unified description via some nice generating function, possibly encoding top intersection degrees of others Schubert cycles, as being investigated in [67]. The notes end with a list of references related with Schubert calculus, certainly not complete, but probably sufficient to let the Reader finding his own path to study this beautiful subject.

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Convention • The grassmannian of k-planes of a vector space V shall

be denoted by Gk (V ). If V = Fn , F any field, one will write Gk (Fn ). When F = C, the notation G(k, n) shall be used instead of Gk (Cn ). • Einstein’s convention summation will be used.

If (v , . . . , v ) are the components of an element v ∈ M with respect to the basis (e1 , . . . , en ), then one will write 1

n

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instead of

v=

n X

v i ei .

i=1

i Similarly, if α ∈ M ∨ , one will write α = αi ǫP instead of Pn n i i j i j i=1 αi ǫ . Also one will write Aj ǫ instead of j=1 Aj ǫ .

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CONTENTS

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Chapter 1

Propaganda The “official” mathematical language, which the author tried to write these notes with, may cause people, who never looked at this subject before, to think that things are more difficult than they really are. To avoid such an undesired feeling, this section aims, thus, to draw an informal and friendly path to show that whatever will be going on, it will in the most natural way. Instead of starting by explaining what a grassmannian is, which is not strictly necessary at this stage, one chooses, for the Reader’s convenience, to enter soon into the core of the subject, by describing the algebraic model which is the main character of this play.

1.1

Revisiting (some) Calculus

1.1.1 Derivatives and Leibniz’s Rules. Believe you or not, the basic manipulation devices proper of Schubert Calculus should already be in the toolkit of any reader who took a basic course in calculus and principles of linear algebra. To start with, and to see why, let C ∞ (R) be the algebra of all real C ∞ -differentiable functions f : R −→ R of one real variable x. The derivative d : C ∞ (R) −→ C ∞ (R), dx 3

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CHAPTER 1. PROPAGANDA

is a linear map f 7→ df /dx, which operates on the product of two functions according to the well known Leibniz’s rule: dg d df (f g) = ·g+f · . dx dx dx

(1.1)

The k th -iterated of d/dx is a linear operator, too, denoted by dk /dxk , which satisfies a Leibniz-like rule, induced by that enjoyed by d/dx. For instance: 2 d dg d d d df (f g) = (f g) = ·g+f · = dx dx dx dx dx dx df dg df dg d2 g d2 f · g + · + · + f · = = dx2 dx dx dx dx dx2 2 2 d f df dg d g = ·g+2 · + f · 2. (1.2) 2 dx dx dx dx For sake of notational brevity, let D0 := 1

and Dn :=

1 dn 1 = D1n . n! dxn n!

In particular, D1 is just the first derivative d/dx, while (1.2) can be rewritten more compactly in the form: D2 (f g) = D2 f · g + D1 f · D1 g + f · D2 g. In general, by simple induction: Dn (f g) =

n X i=0

Di f · Dn−i g.

Again, by induction on the number of factors: X Dn (f1 . . . fk ) = Dn1 (f1 ) · . . . · Dnk (fk ).

(1.3)

(1.4)

n1 +...+nk =n

The Reader will have no difficulty in proving that the following Newton binomial formula holds: n X n n D1h f · D1n−h g. (1.5) D1 (f g) = h h=0

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5

1.1.2 Example. The coefficient of D1i f · D1n−i g in the expansion (1.5) is ` ´ the binomial coefficient ni .

Formula (1.3) shall be referred to as Leibniz’s rule for Dn . If t is an indeterminate over C ∞ (R), then Dt :=

X i≥0

ti Di : C ∞ (R) −→ C ∞ (R)[[t]]

is the Taylor formal power series. The terminology is motivated by the fact that {(Di f )(x0 ), i ≥ 0} is precisely the set of coefficients of the Taylor series associated to the function f in a neighbourhood of the point x0 . Using the formal power series Dt , all the Leibniz’s rules holding for Di , i ≥ 0, can be summarized by the following elegant equality: Dt (f · g) = Dt (f ) · Dt (g), i.e. Dt is a R-algebra homomorphism from C ∞ (R) to C ∞ (R)[[t]]. In general, if A is a commutative B-algebra, any B-algebra homomorphism A −→ A[[t]] is said to be a Hasse-Schmidt derivation (Cf. [54], p. 208). To say it with a slogan, Hasse-Schmidt derivations realize everybody’s student dream: the “derivative” of the product is the product of “derivatives”. 1.1.3 Determinants in C ∞ (R). For 1 ≤ i ≤ k, let ui = (ui1 , . . . , uik ), be a k-tuple of C ∞ real functions. For sake of brevity, denote by u1 ∧ . . . ∧ uk the determinant: u11 u12 . . . u1k u21 u22 . . . u2k (1.6) .. .. .. . .. . . . . uk1 uk2 . . . ukk

The function

u1 ∧ . . . ∧ uk : R −→ R

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is C ∞ , being sums of products of C ∞ functions, and is skew-symmetric with respect to the arguments, namely: uτ (1) ∧ . . . ∧ uτ (k) = (−1)|τ | u1 ∧ . . . ∧ uk , where τ ∈ Sk is a permutation on k elements and |τ | is the number of simple transpositions modulo 2: this follows from the known properties of determinants. Moreover, if ui = uj , for some i < j, then u1 ∧ . . . ∧ uk = 0. We assume known the following formula: D1 (u1 ∧ . . . ∧ uk ) =

k X i=1

u1 ∧ . . . ∧ u′i ∧ . . . ∧ uk ,

(1.7)

where u1 ∧. . .∧u′i ∧. . .∧uk denotes the determinant gotten from (1.6) by differentiating the ith row and keeping untouched all the others. In other words: the operator D1 satisfies Leibniz’s rule with respect to the symbol ∧. The proof of (1.7) merely consists in expanding the determinant as sum of products of the uij ’s and then exploiting the linearity of D1 together with the necessary iterations of Leibniz’s rule. Fantasy now suggests to embed the derivative D1 , with respect to ∧, into a full family of differential operators D := (D1 , D2 , . . .), by imitating formula (1.4), in a perfectly formal way: Dh (u1 ∧ . . . ∧ uk ) =

X

(h1 )

u1

(hk )

∧ . . . ∧ uk

,

(1.8)

h1 +...+hk =h hi ≥0 (h)

where ui

means D1h ui = (D1h ui1 , . . . , D1h uik ).

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1.2. WRONSKIANS AND THEIR DERIVATIVES

1.2

7

Wronskians and their Derivatives

1.2.1 Let f := (f1 , . . . , fk ) be a k-tuple of C ∞ real functions on the real line. The wronskian determinant of f is: f1 f2 ... fk ′ f1 f2′ ... fk′ W (f ) := W (f1 , . . . , fk ) = .. (1.9) .. .. , .. . . . . (k−1) (k−1) (k−1) f f ... f 1

2

k

where f (i) = D1i f . As in Sect. 1.1.3, wronskians shall be shortly denoted as f ∧ f ′ ∧ . . . ∧ f k−1 .

Since D1 behaves obeying Leibniz’s rule with respect to ∧, one has, e.g.: D1 (f ∧ f ′ ∧ . . . ∧ f k−1 ) = f ∧ f ′ ∧ . . . ∧ f (k−2) ∧ f (k) . A generalized wronskian is an expression of the form: f (i1 ) ∧ f (i2 ) ∧ . . . ∧ f (ik−1 ) .

(1.10)

The Reader may easily convince himself that iterated derivatives of “standard” wronskians like (1.9) are integral linear combinations of generalized wronskians. 1.2.2 Remark. What we called generalized wronskian is a particular case of a Schmidt wronskian, used by Schmidt ([72]) to study P Weierstrass points in positive characteristic. If A is a B-algebra and Dt := i≥0 Di ti : A −→ A[[t]] is a Hasse-Schmidt derivation, he defines: ˛ ˛ ˛ Di1 a1 Di1 a2 ... Di1 ak ˛ ˛ ˛ ˛ Di2 a1 Di2 a2 ... Di2 ak ˛ ˛ ˛ ∆(i1 ,...,ik ) (a) = ∆(i1 ,...,ik ) (a1 , ..., ak ) = ˛ . .. .. ˛ , .. ˛ .. . . . ˛˛ ˛ ˛Di a1 Di a2 ... Di ak ˛ k k k (1.11) for each k-tuple a = (a1 , a2 , . . . , ak ), ai ∈ A, and any 1 < i1 < i2 < . . . < ik . What we call, somewhat unproperly, a generalized wronskian is a Schmidth wronskian, where Di = D1i .

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1.2.3 Example. Let f := (f1 , f2 ) ∈ C ∞ (R). Then ˛ ˛ ˛f1 f2 ˛ ˛ = f1 f2′ − f2 f1′ , f ∧ f ′ = ˛˛ ′ f1 f2′ ˛ from which:

D1 (f ∧ f ′ )

=

D1 (f1 f2′ − f2 f1′ ) =

=

f1′ f2′ + f1 f2′′ − f2′ f1′ − f2 f1′′ = ˛ ˛ ˛ f1 f2 ˛ ′′ ˛ f1 f2′′ − f2 f1′′ = ˛˛ ′′ ′′ ˛ = f ∧ f . f1 f2

= Similarly, D1 (f ∧ f ′′ )

=

D1 (f1 f2′′ − f2 f1′′ ) =

=

f1′ f2′′ + f1 f2′′′ − f2′ f1′′ − f2 f1′′′ =

=

f1′ f2′′ − f2′ f1′′ + f1 f2′′′ − f2 f1′′′ = ˛ ′ ˛ ˛ ˛ ˛ f1 f2′ ˛ ˛ f1 ˛ 2 ˛ ˛ ′′ ˛ + ˛ ′′′ f′′′ = f ′ ∧ f ′′ + f ∧ f ′′′ . ′′ ˛f1 f2 ˛ ˛f1 f2 ˛

=

Differentiating once more, one has:

D13 (f ∧ f ′ ) = D1 (f ′ ∧ f ′′ + f ∧ f ′′′ ) = 2f ′ ∧ f ′′′ + f ∧ f (iv) , and then1 D14 (f ∧ f ′ ) = 2 · f ′′ ∧ f ′′′ + 3f ′ ∧ f (iv) + f ∧ f (v) .

(1.12)

1.2.4 Claim. Any Reader able to follow computations of Example 1.2.3 (who did not?) is also able to perform computations in the Chow (or cohomology) intersection ring of the grassmannians G(2, n) (n ≥ 2) (See Sect. 1.4 below), whatever that means. Indeed, all the coefficients of the determinants occurring in the expansion of a derivative of a wronskian, are positive integers having beautiful interpretations in terms of enumerative geometry of projective spaces. For instance, the coefficient 2 multiplying f ′′ ∧ f ′′′ in formula (1.12), has computed 1 It is being implicitly assumed that no generalized wronskian vanishes identically as a function: this is the case, e.g., if f1 , f2 form a basis of the space of solutions of a linear differential equation like y ′′ − (a + b)y ′ + aby = 0, with a 6= b.

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9

the number of lines of the projective 3-space intersecting 4 others s L4 L1

L3

r

L2

in general position. In general, the coefficient of f n−k ∧ f n−k+1 ∧ . . . ∧ f (n−1) occurring in k(n−k) the expansion of D1 (f ∧ . . . ∧ f (k−1 ) is the degree of the Pl¨ ucker image of the complex grassmannian n G(k, n) in P(k )−1 . Explaining why the emphasized sentence holds true2 , is among the purposes of these notes. It basically amounts to prove Theorem 5.4.1 which implies that differentiating a wronskian is the same as applying a Pieri-like formula in the intersection ring of the grassmannian. 1.2.5 The operators Dh defined by formula (1.8) deserve a few more words. Indeed one has: Dh (u(i1 ) ∧ u(i2 ) ∧ . . . ∧ u(ik ) ) = =

X

u(i1 +h1 ) ∧ u(i2 +h2 ) ∧ . . . ∧ u(ik +hk ) .

h1 +...+hk =h hi ≥0

So, for example, Dh (u ∧ u′ ∧ . . . ∧ u(k−1) ) = u ∧ u′ ∧ . . . ∧ u(k−2) ∧ u(k+h−1) . 2 The explanation given in [48] is that the k th exterior power of the polynomial ring in one indeterminate is a free module of rank 1 over the ring of symmetric functions on k-variables, which is in turn isomorphic to the intersection ring of the grassmannian of k-planes.

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1.2.6 The other formal tool, that the Reader should be aware of, is integration by parts to find primitives of functions. For instance, wanting to look for a function f ∈ C ∞ (R) such that D1 f = x cos x, one argues as follows: x cos x = = =

D1 (x sin x) − sin x = D1 (x sin x) − D1 (− cos x) =

D1 (x sin x + cos x + c),

having applied the formula f · D1 g = D1 (f g) − D1 f · g, often written in the calculus textbooks as: Z Z f dg = f g − df · g. One wishes to practise a similar integration by parts on generalized wronskians, with the purpose to climb up to reach the wronskian itself. f ′ ∧ f ′′ = D1 (f ∧ f ′′ ) − f ∧ f ′′′ = = D12 (f ∧ f ′ ) − D2 (f ∧ f ′ ) = (D12 − D2 )(f ∧ f ′ ).

(1.13)

Equation (1.13) can also be written in the form: D D2 f ∧ f ′. f ′ ∧ f ′′ = 1 1 D1

Some healthy experimental mathematics, based on explicitly working out examples, may easily convince the Reader that suitable integrations by parts yield, in general: u(l1 ) ∧ u(1+l2 ) ∧ . . . ∧ u(k−1+lk ) = det(Dlj −j+i ) · u ∧ u′ ∧ . . . ∧ u(k−1) (1.14)

where, by convention, D0 = 1 and Dh = 0 if h < 0. Relation (1.14) is Giambelli’s formula for wronskians or, possibly in spite of the most skeptical Reader, Giambelli’s formula of Schubert calculus!

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Algebraic Schubert Calculus

Now that the Reader believes to possess the main tools for playing with Schubert calculus, here is an attempt to define it in a purely abstract algebraic setting, with no reference at all to the cohomology of grassmannians. It is just an abstract formalization of the yoga of differentiating wronskians. 1.3.1 Let Mn be a free Z-module of rank n spanned by (ǫ1 , . . . , ǫn )3 Vk and, for each k ≥ 2, let Mn be the k th exterior power of Mn (see Section 2.2). The latter is a Z-module spanned by the symbols ǫi1 ∧ . . . ∧ ǫik with 1 ≤ ij ≤ n, 1 ≤ j ≤ k modulo the relations: ǫiτ (1) ∧ . . . ∧ ǫiτ (k) = (−1)|τ | ǫi1 ∧ . . . ∧ ǫik , where τ ∈ Sk is any permutation on k-elements and |τ | is the number i1 ik of its transpositions modulo 2. In particular Vk ǫ ∧. . .∧ǫ = 0 whenever ij il ǫ = ǫ for some j < l. It follows that M n is freely generated by ǫVi1 ∧ . . . ∧ ǫik with 1 ≤ i < . . . < i ≤ n. One sets, by convention, 1 k V 0 Mn = Z and 1 Mn = Mn .

1.3.2 Let D1 : Mn −→ Mn be, now, the endomorphism sending D1 ǫi onto ǫi+1 , if 1 ≤ i ≤ n − 1, and to 0 otherwise. Denote by Di the ith iteration D1i of D1 , and set D10 = idMn . Clearly Di (ǫj ) = 0, if i + j < n and Dj = 0, if j ≥ n. The goal is now to extend the family of endomorphisms D := {Di : Mn −→ Mn }i≥0

V to endomorphisms of k Mn , for all 2 ≤ k ≤ n. Abusing notation, they will be still indicated by Di . The extension will be achieved by imposing Leibniz’s that Di has Vk−1rule with respect to ∧.VSuppose k been extended to Mn . Since any η ∈ Mn can be written V as a finite sum of elements of the form ǫi ∧ αi , αi ∈ k−1 Mn , it

3 The reason why one puts the upper indices is twofold: on one side because it reminds us the “primes” used to denote derivatives, and on the other hand because it will be interpreted, later on, as a basis of a dual vector space, and in these notes we use the Einstein notation, to put lower indices to the element of a basis and the upper indices to the elements of its dual basis.

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is sufficient to define the extension on elements of this form via the equality: h X Dh (ǫi ∧ αi ) = Dj ǫi ∧ Dh−j αi . j=0

Indeed, this makes Dh (h ≥ 0) into an endomorphism of the exterior algebra of Mn : k ^ M^ Mn := Mn k≥0

V0

V1 where, by definition, Mn ∼ Mn ∼ = Z and = Mn . ThePsequence i D = {Di }i≥0 , as well as the formal power series Dt = i≥0 Di t associated to it, will be named Schubert derivation (S-derivation) (Cf. Sect. 5.3). 1.3.3 Example. Compute: D2 (ǫ2 ∧ ǫ4 ∧ ǫ6 ) One has: D2 (ǫ2 ∧ ǫ4 ∧ ǫ6 ) = D2 (ǫ2 ) ∧ ǫ4 ∧ ǫ6 + +D1 (ǫ2 ) ∧ D1 (ǫ4 ∧ ǫ6 ) + ǫ2 ∧ D2 (ǫ4 ∧ ǫ6 ) = The first summand vanishes, since D2 (ǫ2 ) = ǫ4 and ǫ4 ∧ ǫ4 = 0, hence D2 (ǫ2 ∧ ǫ4 ∧ ǫ6 ) = ǫ3 ∧ (D1 ǫ4 ∧ ǫ6 + ǫ4 ∧ D1 ǫ6 ) + ǫ2 ∧ D2 (ǫ4 ∧ ǫ6 ) =

ǫ3 ∧ ǫ5 ∧ ǫ6 + ǫ3 ∧ ǫ4 ∧ ǫ7 + ǫ2 ∧ ǫ5 ∧ ǫ7 + ǫ2 ∧ ǫ4 ∧ ǫ8 .

1.3.4 All the matter consists, then, in differentiating and cancelling terms with opposit sign. However, the practice of many examples suggests how to predict the terms surviving after vanishing and cancelations. This is the content of Pieri’s formula for S-derivations (Theorem 5.4.1): X Dh (ǫi1 ∧ . . . ∧ ǫik ) = ǫi1 +h1 ∧ . . . ∧ ǫik +hk , (hi )

the sum over all (h1 , . . . , hk ) such that

P

hi = h and

i1 ≤ i1 + h1 < i2 ≤ . . . ≤ ik−1 + hk−1 < ik .

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To each strictly increasing sequence I := (i1 , . . . , ik ) of k positive integers, said to be a k-schindex in Section 2.1.5 , one may associate a partition P I(λ) = (rk , . . . , r1 ), where rj = ij − j. The weight of λ is |λ| = ri . If one writes ǫλ instead of ǫ1+r1 ∧ . . . ∧ ǫk+rk , then Pieri’s formula translates into: X Dh ǫλ = ǫµ , µ

wherePthe sum P is over all partitions µ = (sk ≥ . . . ≥ s1 ≥ 0) such that si = ri + h and: 0 ≤ r1 ≤ s1 ≤ r2 ≤ s2 ≤ . . . ≤ sh−1 ≤ rh ≤ sh ,

(1.15)

coinciding with the combinatorial Pieri’s formula: indeed, the Young diagram (Sect. 4.3) Y (µ) of µ satisfying (1.15) is gotten by adding i boxes to Y (λ) in all possible ways, not two on the same column. Furthermore, everybody’s basic calculus experience (Cf. Sect. 1.2.6) Vk proves that, for any ǫi1 ∧. . .∧ǫik ∈ Mn , there exists a Z-polynomial expression Gi1 ...ik (D) in the Di ’s such that ǫi1 ∧ . . . ∧ ǫik = Gi1 ...ik (D) · ǫ1 ∧ . . . ∧ ǫk . The above fact shall be formally proven in Section 5.5, Theorem 5.5.9. Useless to say, the proof consists in using integration by parts and it is nothing more than a mere generalization and formalization of the following example: ǫ2 ∧ ǫ3

= = = =

D1 (ǫ1 ∧ ǫ3 ) − ǫ1 ∧ D1 ǫ3 = D1 (ǫ1 ∧ ǫ3 ) − ǫ1 ∧ ǫ4 =

D12 (ǫ1 ∧ ǫ2 ) − D2 (ǫ1 ∧ ǫ2 ) = (D12 − D2 )(ǫ1 ∧ ǫ2 )

already encountered in “wronskian shape”. V 1.3.5 Exercise. Let ǫ2 ∧ǫ5 ∧ǫ6 ∈ 3 Mn , n ≥ 6. Find a polynomial expression G256 (D) in D1 , D2 , D3 such that ǫ2 ∧ ǫ5 ∧ ǫ6 = G256 (D) · ǫ1 ∧ ǫ2 ∧ ǫ3 . (Hint: I would start so: ǫ2 ∧ǫ5 ∧ǫ6 = D1 (ǫ1 ∧ǫ5 ∧ǫ6 )−ǫ2 ∧D1 (ǫ5 ∧ǫ6 ) = . . .)

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The general philosophy is that one does not need to know the explicit form of Gi1 ...ik : everybody just computes its own expression4 . The practise of many computations then develops the feeling for what should be the most rapid way to get the answer. It is however natural to apply the same “integration” procedure to figure out a canonical shape for the polynomial Gi1 ...ik . This will be done in Sect. 5.8, where Giambelli’s formula will be proven as a corollary of a more general expression holding in the exterior algebra of the module M (Theorem (5.8.1)). 1.3.6 To believe in the existence of Gi1 ...ik (D) such that Gi1 ...ik (D)ǫ1 ∧ . . . ∧ ǫk = ǫi1 ∧ . . . ∧ ǫik means to believe that the map P (D) 7→ P (D)ǫ1 ∧ . . . ∧ ǫk is surjective. This allows to define a product structure on

Vk

M:

(ǫi1 ∧ . . . ∧ ǫik ) · (ǫj1 ∧ . . . ∧ ǫjk ) = (Gi1 ...ik (D) · Gj1 ...jk (D))ǫ1 ∧ . . . ∧ ǫk which is indeed isomorphic to the intersection product in the Chow group of the grassmannian G(k, n). 1.3.7 To end this section it seems worth to remark that formula (1.5) holds verbatim up to replacing · with ∧: n X n n D1 (α ∧ β) = D1i α ∧ D1n−i β. i i=0 The easy (and probably not necessary) proof is given in Sect. 6.3.1. Observe now that X D1h (ǫ1 ∧ ǫ2 ) = αij · ǫi ∧ ǫj , 0≤i<j

i+j=h+3

with αij ∈ Z to be determined. This is obvious by the definition of D1 . It follows that ǫn−1 ∧ ǫn can only occur in the expression of 2(n−2) 1 D1 (ǫ ∧ ǫ2 ). 4 There are more than one and all equivalent modulo the ideal of relations of the cohomology ring of the grassmannian (See Sect. 5.5)

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1.3.8 Example (Analogous to 1.1.2). One aims to compute the co2(n−2) 1 efficient of ǫn−1 ∧ ǫn in the expansion of D1 ǫ ∧ ǫ2 . Notice that such a coefficient gets the contribution of that multiplying ǫn−1 ∧ ǫn as well as that multiplying ǫn ∧ ǫn−1 . As a matter of fact one has: ! ! 2(n − 2) n−1 2(n − 2) n 2(n−2) 1 2 n D1 (ǫ ∧ ǫ )) = ǫ ∧ǫ + ǫ ∧ ǫn−1 + o.t. n−2 n−1 where o.t. means all the remaining summands. Therefore, the sought for coefficient is: ! ! 2(n − 2) 2(n − 2) (2(n − 2))! d2,n = − = . n−2 n−1 (n − 2)!(n − 1)! See example 6.3.3.

Later on, it will be shown that this example is just computing the degree of the Pl¨ ucker image of the grassmannian G(2, n) Or, alternatively, the number of lines of Pn−1 meeting 2(n − 2) linear subspaces of dimension n − 3 in general position.

1.4

Grassmannians

1.4.1 To deal with enumerative problems such as that of Example 1.3.8, one needs to introduce grassmannians. A grassmannian is primarily a linear algebraic object: if V is a vector space over any field F, denote by Gk (V ) the set of all vector subspaces of V of dimension k. What can be said about Gk (V )? If k = 0, then G0 (V ) = {0V } and if dimF V = n, then Gn (V ) = V . Since any inclusion W1 ⊆ W2 of vector spaces implies the inequality dim(W1 ) ≤ dim(W2 ), one sees that Gk (V ) = ∅ whenever k is bigger than the dimension of V , when finite. If dim(V ) = n ≤ 1, there is not much to say: G0 (V ) and G1 (V ) consist of at most 1 element. One may think, at first, that even for higher dimensional vector spaces, the situation is not that interesting. Indeed, all the k-dimensional subspaces of V look the same in at least two different senses. The former is that any two vector spaces over the same field and of the same dimension are abstractly

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isomorphic. The second is that, given any two k-planes of V , there always exists an element of the general linear group Gl(V ) sending one into the other, as it will be proved in Proposition (4.1.1). 1.4.2 Another way to look at grassmannians is to think of them as the set of all k-dimensional linear subspaces of a projective space Pn . A linear subspace of a projective space P(V ) is the projective space associated to a vector subspace W ⊆ V . So, for instance, the grassmannian G2 (R4 ) of 2-planes of R4 can be regarded as the grassmannian of projective lines of P3 (R). There is an obvious bijection of sets: G1+k (V ) = G(k, P(V )). In particular, P(V ) = G1 (V ) = G(0, P(V )), i.e. the set of lines of V is the same as the set of points of P(V ). 1.4.3 Example. The set G1 (R2 ) can be thought of as the set of all the lines passing through the origin of the standard (x, y)-plane of high school analytic geometry. y

x

A pencil of lines is a model for the projective line P1 Any such line can be written in the form ax + by = 0, where (a, b) is a non-zero pair of R2 , uniquely determined up to a non-zero multiple. The set G1 (R2 ) is called the projective R-line. It turns out that P1 (R) := G1 (R2 ) is a compact connected differentiable manifold of dimension 1, diffeomorphic to the circle S 1 := {(x, y) ∈ R2 | x2 + y 2 = 1}, with the differentiable structure induced by the stereographic projections.

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N

x

S

Stereographic projections: from the “north” and “south” poles Similarly, if F = C, then P1 (C) := G1 (C2 ) is a compact connected Riemann Surface bi-holomorphically equivalent to the Riemann Sphere S 2 (see e.g. [10]).

1.4.4 For the limited purposes of these notes, even the field R is not enough to keep the exposition as simple as possible, without loosing elegance. Readers are invited to feel the reason through the following example, whose main character is the quadric hyperfurface x0 x5 − x1 x4 + x2 x3 = 0 of P5 (R), the real 5-dimensional projective space with its homogeneous coordinates [x0 , x1 , . . . , x5 ]. Consider the line [v, 0, u, u, au + bv, v], where [u, v] ∈ P1 and (0, 0) 6= (a, b) ∈ R2 . Where do they intersect? Trying to solve the system one finds: u2 + v 2 = 0 which has no solution over the reals, while the line [v, 0, u, −u, au + bv, v] intersects the given quadric at two points. To avoid such a dichotomy, one wishes that any general line intersects a quadric surface at two distinct points. Allowing the coefficients to run over an algebraically closed field, and ours will be C, one is led to work with complex grassmannians, parametrizing k-dimensional subspaces of a complex vector space V . The reason is that the intersection of a line in P5

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with a quadric hypersurface, counts indeed the number of lines in P3 intersecting four others in general position. For this reason, in the rest of such introduction, all the vector spaces will be thought over the complex field. 1.4.5 The fact that any two k-planes are indistinguishable in the sense of Proposition 4.1.1 cries for the introduction of some kind of “landmark”. For instance, picking an h-plane Π of V , a general kplane will intersect it in a subspace of dimension n − h − k: general means, here, out of the space of solutions of some linear system of equations. However, they may be k-planes in special position intersecting Π in a subspace of dimension bigger than expected : if h < k, this is the case, for instance, of any k-plane containing Π! In this sense Π discriminates planes having an expected behaviour from those that are in special position. One may refine the landamrk : for instance one can pick two vector subspaces W1 and W2 , of dimension h1 and h2 respectively, and then distinguish the k-planes into: i) those which are in general position with respect to W1 and W2 , ii) those which are in general position with respect to one but not with respect to the other; iii) and those which are in special position with respect to both. One soon realizes that the finest description is achieved by refining such kinds of “reference systems” as much as possible. This naturally leads to the important notion of a flag of subspaces of V , introduced in Section 4.2 1.4.6 It turns out that the complex grassmannian Gk (V ) (or her projective sister Gk−1 (P(V )), is a complete smooth connected algebraic variety. As a geometrical object it is interesting in its own, either from the point of view of differential geometry (see e.g. [41], [11]) or from the point of view of algebraic geometry. For the purposes of this introductory chapter it is enough for us to recall that very natural bundles (i.e. algebraic varieties which locally look like the product of a variety and a vector space) live on Gk (V ). Indeed, there is a tautological exact sequence: 0 −→ Tk −→ Gk (V ) × V −→ Qk −→ 0 where Tk := {([Λ], v) ∈ Gk (V ) × V | v ∈ [Λ]}

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is said to be the tautological bundle: the fiber of the projection π : Tk −→ Gk (V ) over [Λ] ∈ Gk (V ) is the k-plane [Λ] itself5 . As for the universal quotient bundle Qk , its fiber over [Λ] is the quotient vector space V /[Λ]. A linear form φ on V induces a bundle homomorphism: φ : Tk −→ Gk (V ) × C, defined by φ([Λ], v) = ([Λ], φ(v)), which will be very important for us. In fact the operators Di quickly introduced in Section 1.3.2 can be identified with the Chern classes of the universal quotient bundle: they are the so calles special Schubert cycles on Gk (V ).

5 The

reason for brackets around Λ will be clear in Chapter 4.

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Chapter 2

Preliminaries The purpose of this section is to render the notes as self contained as possible. Most of the readers can skip this part or use it as a list of terms he may get better acquainted with in many good available references.

2.1

Combinatorics

2.1.1 Partitions. A partition is a non-increasing sequence λ = (λ1 , λ2 , . . .) of non-negative integers such that all but finitely many terms are 0. Partitions will be denoted with underlined greek letters (e.g. λ, µ, ν). Let P be the set of all partitions. The length of λ ∈ P is the number of its parts: ℓ(λ) = ♯{i ≥ 1 | λi 6= 0}. while its weight is |λ| =

X

λi .

i≥1

One denotes by Pn the set of all partitions of the integer n, i.e.: Pn := {λ ∈ P | |λ| = n}. 20

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If λ ∈ Pn , then 1 ≤ ℓ(λ) ≤ n. The partition of minimum length is (n) while that of maximum length is (1, 1, . . . , 1). A partition | {z } n times

of length k will be usually identified with the finite non-increasing sequence of its non-zero terms (λ1 , . . . , λk ). Let: ℓ(k) := {λ ∈ Nk | λ is a partition of length ≤ k}. Then λ ∈ ℓ(k) is a k-tuple of non increasing integers (some of them may be zero). 2.1.2 Example. The elements of the set P5 of the partitions of the integer 5 are: (5),

(41),

(32),

(311),

(221),

(2111),

(11111),

where it has been omitted the infinite sequence of final zeroes.

2.1.3 Historical Remark. Let p(n) be the cardinality of the set Pn of the partitions of the integer n. By convention p(0) = 1. Below is a table of the values of p(n) for n ≤ 10. n p(n)

0 1

1 1

2 2

3 3

4 5

5 7

6 11

7 15

8 22

9 30

10 42

A nice formula by Hardy and Ramanujan, see [55], shows that the function p(n) is asymptotic to p exp(π 2n/3) √ 4n 3 as n → ∞. So, for instance, p(100) ∼ 1.9 · 108 . 2.1.4 There are several way to represent a partition. a) Reverse Order. For each 1 ≤ i ≤ k define: ri : ℓ(k) −→ N, by ri (λ) = λk+1−i . The k-tuple r λ = (r1 (λ), . . . , rk (λ)) is λ in the reverse order. Most of the times, λ will be skipped from the notation and one will shortly write ri instead of ri (λ). The partition λ will be simply written as (rk , . . . , r1 ).

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CHAPTER 2. PRELIMINARIES

b) Young Diagrams. To each partition λ ∈ P, one may associate its Young diagram Y (λ). The diagram Y (λ) is an array with k-rows, the rows are left-aligned boxes such that the i-th rows has λi boxes. The Young diagram of the partition (53111) is depicted below.

More formally: Y (λ) := {(i, j) ∈ Z2 | 1 ≤ j ≤ λi }. The length of a partition is the number of rows of its Young diagram while its weight is the number of boxes. c) Multiplicities. Define mi (λ) = ♯{λj | λj = i}. The integer mi (λ) is the multiplicity which the integer i occurs with in the partition λ. Then a partition can be also written as λ = (1m1 2m2 . . . hmh ). In this case one has: ℓ(λ) = m1 + . . . + mh

and

|λ| = m1 + 2m2 + . . . + hmh .

The conjugated of a partition λ is the partition λ′ whose Young diagram Y (λ′ ) is the transpose of Y (λ). For example, if λ = (53111), then λ = (52211) with Young diagram depicted below

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2.1.5 Schindices. Let I k = {(i1 , . . . , ik ) ∈ Nk | 1 ≤ i1 < i2 < . . . < ik } and Ink = {I ∈ I k | ik ≤ n}. In other words, I k is the set of all increasing ordered k-tuples of positive integers. Any such k-tuple will be called, for comodity, a kschindex, as a contraction of Schubert and index. The reason for this name is that for each k-schindex I there is a grassmannian Gk (V ) with a Schubert variety having precisely that Schubert index. Let I = ∪k≥0 I k . The set I is said to be the set of schindices. Any I ∈ I is a k-schindex for some k. Schindices will be denoted by capital roman letters I, J, K, . . .. 2.1.6 Schindices and Partitions are strictly related. To each I ∈ I k , one may associate the partition λ(I) defined by: λ(I) = (ik − k, . . . , i1 − 1). Conversely, if λ = (rk , . . . , r1 ) ∈ ℓ(k), one sets: I(λ) = (1 + r1 , . . . , k + rk ). Clearly λ : I k −→ P and I : P −→ I k are bijections inverse one of each other. The notation for such bijections is clearly abused. The length of a k-schindex is k, while its weight is, by definition, wt(I) = |λ(I)| =

k X i=1

ri = (i1 − 1) + . . . + (ik − k).

On the set I k we shall put two different orders: a) The Bruhat-Chevalley order: (i1 , . . . , ik ) (j1 , . . . , jk ) ⇐⇒ ip ≤ jp ,

∀ 1 ≤ p ≤ k,

writing (i1 , . . . , ik ) ≺ (j1 , . . . , jk ) if (i1 , . . . , ik ) (j1 , . . . , jk ) and (i1 , . . . , ik ) 6= (j1 , . . . , jk ).

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b) The lexicographic order: (i1 , . . . , ik ) (j1 , . . . , jk ) if and only if either ip = jp for all 1 ≤ p ≤ k or if q := min{p | ip 6= jp } ⇒ iq < jq . 2.1.7 Example. The set I42 = {(i1 , i2 ) ∈ I 2 | i2 ≤ 4} can be ordered be ordered, according to the Bruhat order, as follows: (1, 2) (1, 3) (2, 3)

(1, 4) (2, 4) (3, 4)

while in the lexicographic order: (1, 2) ≺ (1, 3) ≺ (1, 4) ≺ (2, 3) ≺ (2, 4) ≺ (3, 4). 2.1.8 The ring Z[T]. Let T = (T1 , T2 , . . .) be an infinite sequence of indeterminates over Z and denote by Z[T] the ring Z[T1 , T2 , . . .] of polynomials in the indeterminates T with Z-coefficients. For each partition λ = (1m1 . . . hmh ), define the monomial Tλ by: Tλ = T1m1 · . . . · Thmh . Then Z[T] admits a direct sum decomposition of submodules: M Z[T] = Z[T]h , h≥0

where Z[T]h =

L

Z[T]0 = Z;

|λ|=h Z

· Tλ . For example:

Z[T]1 = Z · T1 ;

Z[T]2 = Z · T12 ⊕ Z · T2 .

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One defines the degree of a monomial Tλ as the weight of the partition λ. Then deg(Ti ) = i, for all i ≥ 0 (where by convention T0 = 1). The ring Z[T1 , . . . , Tn ] of polynomials in n-indeterminates is a quotient of Z[T]. 2.1.9 Let A be a commutative ring with unit and let c : N −→ A be a sequence. Set ci = c(i). To any partition λ = (rk , rk−1 , . . . , r1 ) of length k, let ∆λ (c) ∈ A be defined as: cr 1 cr2 +1 . . . crk +k−1 cr1 −1 cr 2 . . . crk +k−2 ∆λ (c) = . (2.1) = det(crj +j−i ). . .. .. .. .. . . cr −k cr −k+1 . . . crk 1 2

setting ci = 0 if i < 0. Let I(λ) = I = (i1 , . . . , ik ) be the k-schindex associated to the partition λ. Then (2.1) can be equivalently written as: ci1 −1 ci2 −1 . . . cik −1 ci1 −2 ci2 −2 . . . cik −2 ∆I (c) := ∆(i1 ...ik ) (c) = . (2.2) .. .. . .. .. . . . ci −k ci −k . . . ci −k 1 2 k

Both equations (2.1) and (2.2) are said to be Giambelli’s determinant of c associated to the partition λ or to the k-schindex I. If A := Z[T] as above, one writes ∆λ (T) ∈ Z[T] (or ∆I (T)) for the corresponding Giambelli’s determinants associated to the sequence T = (T1 , T2 , . . .), i.e. ∆λ (T) = det(Trj +j−i ).

2.2

Exterior Algebra of a Free Module

2.2.1 To any module M over a commutative ring A, one may associate an algebra (T (M ), ⊗), said to be V tensor algebra (see e.g. [2]). The exterior (or Grassmann) algebra M of M is a suitable quotient of T (M ). For the limited purposes of this exposition, it is sufficient to look at free modules over an integral domain A of characteristic 01 1 Vector

spaces, among them.

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and, in this case, we shall give a practical ad hoc definition of exterior algebra, ready for use. Let M be a free A-module of infinite countable rank, generated by E = (ǫ1 , ǫ2 , . . .). V V 2.2.2 Definition. Define 0 M = A, 1 M = M and, for all Vk k ≥ 2, let M be the A-module generated by all the expressions of the form ǫi1 ∧ . . . ∧ ǫik modulo the relations: ǫiτ (1) ∧ . . . ∧ ǫiτ (k) = (−1)|τ | · ǫi1 ∧ . . . ∧ ǫik . for all τ ∈ Sk . For all k ≥ 0, the A-module k th exterior power of M .

Vk

(2.3)

M is said to be the

Since the characteristic of the ring is 0 (hence 6= 2), relation (2.3) implies that ǫi1 ∧ . . . ∧ ǫik = 0 whenever ij1 = ij2 for some j1 < j2 . It Vk follows that a basis of M is given by k ^

E = {ǫi1 ∧ . . . ∧ ǫik : 1 ≤ i1 < . . . < ik },

the elements being ordered lexicographically. One has clearly a map h ^

M×

k ^

M −→

h+k ^

M.

which is the unique A-bilinear extension of that sending (ǫi1 ∧ . . . ∧ ǫih , ǫi1 ∧ . . . ∧ ǫik ) 7→ ǫi1 ∧ . . . ∧ ǫih ∧ ǫi1 ∧ . . . ∧ ǫik . (2.4)

V 2.2.3 Definition. The exterior algebra of M is the pair ( M, ∧), where k 2 ^ M^ ^ M := M = A ⊕ M ⊕ M ⊕ ... k≥0

and the product ∧ is defined according to formula (2.4). V V Any k M will be thought of as an A-submodule of M . If α ∈ Vk V M ⊂ M , then α is said to be homogeneous of degree k. Any Vk Vk α∈ M is a finite A-linear combination of elements of E. Any

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V β ∈ M is a finite sum of homogeneous elements. There is an Vk obvious map M × . . . × M −→ M defined by {z } | k times

(m1 , . . . , mk ) 7→ m1 ∧ . . . ∧ mk .

Such a map is neither injective (any m ∈ M k with at least two equal components is mapped to 0) nor surjective, as it can be shown on the basis of easy examples: the image is said to be the set of decomposable elements. 2.2.4 Exterior Algebras of Finite Modules. The same construction holds verbatim for the V free submodule Mn of M spanned k by En = (ǫ1 , . . . , ǫn ). A basis for Mn is then k ^

En = {ǫi1 ∧ . . . ∧ ǫik ∈

k ^

E | ik ≤ n}.

V Clearly one has k Mn = 0 if k > n, because in this case any ǫi1 ∧ . . . ∧ ǫik must have two equal indices. One has: k ^ n rkA ( Mn ) = ♯(Ink ) = k

Therefore ^

Mn =

k M^

Mn = A ⊕ Mn ⊕

2 ^

Mn ⊕ . . . ⊕

is itself a free module of finite rank equal to ! k n X ^ X n rkA Mn = = 2n . k k≥0

n ^

Mn

k=0

2.2.5 Example. Let M4 = Zǫ1 ⊕ Zǫ2 ⊕ Zǫ3 ⊕ Zǫ4 . Then a basis of with lexicographic ordering, is: (ǫ1 ∧ ǫ2 , ǫ1 ∧ ǫ3 , ǫ1 ∧ ǫ4 , ǫ2 ∧ ǫ3 , ǫ2 ∧ ǫ4 , ǫ3 ∧ ǫ4 ). V Then rkZ (M ) = 6. Similarly, a basis of 3 M4 is

V2

M,

(ǫ1 ∧ ǫ2 ∧ ǫ3 , ǫ1 ∧ ǫ2 ∧ ǫ4 , ǫ1 ∧ ǫ3 ∧ ǫ4 , ǫ2 ∧ ǫ3 ∧ ǫ4 ). V Then the rank of 3 M4 is 4.

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2.2.6 Example. Let φ : V −→V W be of F-vector V a homomorphism V spaces. The map φ induces a map k φ : k V −→ k W defined as (

k ^

φ)(v1 ∧ . . . ∧ vk ) = φ(v1 ) ∧ . . . ∧ φ(vk ).

V V One checks that rk(φ) = V k if k φ 6= 0 and k+1 φ = 0. If φ ∈ EndF (V ) and dim(V ) = n, then n φ(e1 ∧ . . . ∧ en ) = a · e1 ∧ . . . ∧ en , where En = (e1 , . . . , en ) is a basis of V . The scalar a ∈ F is said to be the determinant of the map φ.

2.3

Review of Intersection Theory

Schubert calculus for grassmannians is a formalism enabling to perform explicit computations in the Chow intersection ring (or integral Cohomology Ring) of a complex grassmannian variety – a smooth algebraic projective variety. The aim of this section is then to quickly collect a few notions intended to describe the idea of Chow ring of a smooth complex projective variety. Universal references for the section below are [16] and [17]. The brazilian reader should also look at [78]. 2.3.1 Order Along a Subvariety. Let X be a smooth complex projective variety of dimension n and let OX be its structure sheaf. The function field k(X) of X is the quotient field of the integral domain OX (U ), for some affine open subset U of X. The definition makes sense because if V is another affine open set of X, then the quotients fields of OX (U ) and OX (V ) are isomorphic. If r ∈ k(X), it can be written as a/b, with a ∈ OX (U ) and b ∈ OX (U )∗ := OX (U ) \ {0}, for some affine open set U . If a ∈ OX (U ) and Y is a (n − 1)dimensional subvariety of X such that Y ∩ U 6= ∅, write: AY (U ) ordY (a) = lAY (U) , pAY (U ) where AY (U ) denotes the localization of OX (U ) at the prime p defining Y ∩ U in U and lA (M ) denotes the lenght of an A-module M (Cf. [16], p. 8 and p. 407). If r ∈ k(X)∗ , one sets: ordY (r) = ordY (a) − ordY (b).

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2.3.2 Cycles. The group Zk (X) of k-cycles of the variety X is the free Z-module generated by all the irreducible subvarieties of X of dimension k. The group of cycles of X is Z∗ (X) := Z0 (X) ⊕ Z1 (X) ⊕ . . . ⊕ Zn (X). If V is a subvariety of X of dimension k one shall denote by [V ] the corresponding k-cycle. To each r ∈ k(X)∗ one may associate the cycle X div(r) := ordY (r)[Y ]. [Y ]∈Zn−1 (X)

One can prove that the above sum is finite (Cf. [16], p. 432). 2.3.3 Rational Equivalence. One says that V ∈ Zk (X) is rationally equivalent to 0 (V ∼ 0) if there are (k + 1)-dimensional subvarieties W1 , . . . , Wj and ri ∈ k(Wi )∗ such that X [V ] = div(ri ).

The k-cycles rationally equivalent to zero form a Z-submodule Ratk (X) of Zk (X). The quotient: Ak (X) :=

Zk (X) Ratk (X)

is said to be the Chow group of k-dimensional cycles modulo rational equivalence. The Chow group (of cycles modulo rational equivalence) of X is: M A∗ (X) := Ai (X). i≥0

In particular, the group An (X) is a free Z-module of rank 1 generated by [X] ∈ Zn (X) = An (X), said to be the fundamental class of X. Each proper morphism f : X −→ Y of C-schemes induces a Zmodule homomorphism: f∗ : Zk (X) −→ Zk (Y ), defined as follows. If [V ] is a generator of Zk (X), then:  if dim(V ) = dim(f (V ) deg(f|V ) · [f (V )] f∗ [V ] = ,  0 if dim(f (V )) < dim(V )

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where by deg(f|V ) we mean the degree of the algebraic field extension [K(V ) : K(f (V ))]. Moreover, if f : X −→ Y is a flat morphism of relative dimension m, one may define a pull-back map f ∗ at the level of the cycles. It is defined as: f∗

: Zk (Y ) −→ Zk+m (X) [V ] 7→ f ∗ ([V ]) = [f −1 (V )].

Now, let f : X −→ Y be a proper morphism. If α ∈ Ratk (X), then f∗ (α) ∈ Ratk (Y ), while if f is flat of relative dimension m and α ∈ Ratk (Y ), then f ∗ α ∈ Ratk+m (X). This implies that f∗ and f ∗ define two homomorphisms of graded Z-modules: f∗ f

∗

: A∗ (X) −→ A∗ (Y )

and

: A∗ (Y ) −→ A∗ (X).

(2.5) (2.6)

Such a fact is proven in [16], Ch. 1. The homomorphisms (2.5) and (2.6) shall be called respectively the proper push-forward (under the assumption that f is proper) and the flat pull-back (under the assumption that f is flat). From now on, if V is a k-dimensional subvariety, the symbol [V ] will denote its Chow class, i.e. the k-cycle determined by it, modulo rational equivalence. Let now W be a k-dimensional scheme. Let Wi be its irreducible components. Then the local rings OW,Wi are artinian local rings ([2]). Define the multiplicity as the OW,Wi -lenght of the module OW,Wi itself. Set: mi = ℓOW,Wi (OW,Wi ). 2.3.4 Definition. The fundamental class of a k-dimensional subscheme is defined to be the Chow class: [W ] =

X i

mi [Wi ] ∈ Ak (X).

Notice that if X is irreducible and reduced then the fundamental class of W is simply the rational equivalence class of the cycle [W ]. Another very important definition, which we already used more than once, without setting it, is that of the degree of a cycle.

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2.3.5 Definition. Let X be a proper scheme of finite type over Spec(F), F being any algebraically closed field. The degree homomorphism: Z X

: A∗ (X) −→ Z,

is defined as: Z

α= X

P  P nP [k(P ) : F] if 

0

if

α = nP [P ] ∈ A0 (X) α ∈ Ai (X), i > 0.

In the above formula [k(P ) : F] denotes the algebraic degree extension of F by the field of the point P , k(P ) = OP /mP , mP being the maximal ideal of the local ring OP . Clearly [k(P ) : F] = 1 when F (as in our case) is algebraically closed. If f : X −→ Y is a morphism of schemes proper over Spec(F), then: Z Z α= f∗ (α). X

Y

2.3.6 Example. Let An be the affine n-space. Then An (An ) = Z[An ]. n We claim that P A0 (A ) = 0: in fact any point in the support of a 0cycle c = P nP P belongs to a line, and on a line any P 0-cycle is the set of poles and zeros of a rational function. Therefore np P = div(rP ) where rP is some rational function on some line passing through P , and then equivalent to 0. From the fact that the obvious projection maps p : An −→ An−k forgetting the first k coordinates induces a surjective map p∗ : Ai (An−k ) −→ Ai+k (An−k ) (Cf. [16], p. 22), it follows that: Ai (An ) = p∗ (A0 (An−i )) = 0.

2.3.7 The Chow Ring. One defines: Ak (X) := An−k (X), the Chow group of cycles of X of codimension k. There is an obvious Z-module isomorphism between A∗ (X) := ⊕Ai (X) and A∗ (X). If X is smooth (as in our hypothesis), one can put on A∗ (X) ∼ = A∗ (X) an intersection product · : Ai (X) × Aj (X) −→ Ai+j (X) , (α, β) 7−→ α·β

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making it into a ring. Such a product has a geometrical interepretation. Recall that two subvarieties V1 and V2 of X intersect properly if and only if codim(V1 ∩ V2 ) = codim(V1 ) + codim(V2 ). Moreover they intersect transversally along W , if ∃w ∈ W such that Tw V1 + Tw V2 = Tw X: the point w is then non singular for W and one has Tw W = Tw V1 ∩ Tw V2 . If V1 and V2 intersect properly, X [V1 ] · [V2 ] = [V1 ∩ V2 ] = mW [W ] (2.7) W

where the sum is over all the irreducible subvarieties W of the scheme theoretical intersection V1 ∩V2 and mW is the intersection multiplicity of V1 and V2 along W . The intersection multiplicity is defined in such a way that if V1 and V2 intersect transversally along W , then mW = 1. This is the reason why the product above is also called intersection product. 2.3.8 Example. If A0 (X) ∼ = Z, generated by the class [pt] of a point (like, e.g., in the case of grassmannians) codim(V1 ) + codim(V2 ) = dim(X), and V1 and V2 intersect transversally, then [V1 ] ∪ [V2 ] = ♯(V1 ∩ V2 )[pt]. However, the number on the right hand side does not depend on V1 and V2 but only on their classes modulo rational equivalence.

2.3.9 Intersection product on homogeneous varieties. If X is transitively acted (say) on the left by a connected algebraic group G (i.e. X is a G-homogeneous variety), an important theorem of Kleiman ([37]) ensures that if [Y1 ] and [Y2 ] are any two cycles, then there exists a Zariski dense open set U ⊂ G such that for each g ∈ U , Y1 and Lg (Y2 ) meet properly, where Lg : X −→ X denotes the left translation. In this case, according to (2.7), [Y1 ] · [gY2 ] = [Y1 ∩ Lg (Y2 )]. Moreover, if the group is rational, e.g. G = Gln (C) as in the case of grassmannians, then the map (Lg ) is proper and (Lg )∗ : A∗ (X) −→ A∗ (X)

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is the identity (Cf. [16], p. 207). Then, in particular, the self intersection [Y ]2 is represented by the intersection of Y with a general translate of it. The conditions above are met for grassmannians varieties (See Proposition 4.1.1). 2.3.10 The Chow group as a module over the Chow ring. The Chow group A∗ (X) inherits a structure of module over A∗ (X) with respect to the cap product: ∩ : A∗ (X) × A∗ (X) −→ A∗ (X) (α, [V ]) 7−→ α ∩ [V ]. Poincaré duality says that the map α 7→ α ∩ [X] induces an isomorphism between Ai (X) ⊗Z Q and An−i (X) ⊗Z Q. The variety X being smooth, the above isomorphism holds indeed over the integers. It is not easy, in general, to compute the Chow ring even of a nice smooth complex variety. 2.3.11 Suppose that U ⊆ X is a Zariski open set and let Y := X \U be its (closed) complement in X. Let i : U −→ X and j : Y −→ X be the inclusions. Then the following exact sequence holds j∗ i∗ Ai (Y )−→Ai (X)−→Ai (U ) −→ 0.

(2.8)

The proof consists first in showing that the exact sequence holds at the cycle level: Zi (Y ) −→ Zi (X) −→ Zi (U ) −→ 0, and then that the above maps pass to rational equivalence. See [16]. 2.3.12 Example. One has that An (Pn ) = Z[Pn ] and Ai (Pn ) = 0 for i > n. If 0 < r < n, Ar (Pn ) is generated by the class of a linear subspace H (r) ∼ = Pr of dimension r. In fact exact sequence (2.8) gives: An−1 (H (1) ) −→ An−1 (Pn ) −→ An−1 (Pn \ H (1) ) −→ 0 and this proves that An−1 (Pn ) is generated by H (1) . Similarly one sees that An−i (Pn ) is generated by An−i (Pn−1 ) which, by induction, is generated by a linear space of dimension n − i.

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2.3.13 Vector Bundles. A holomorphic vector bundle of rank r over X is a complex algebraic variety V together with a surjective morphism π : V −→ X, such that the map π is locally trivial (in the holomorphic or algebraic category). The latter means that there is a family (U, ψU ), where {U } is an open covering of X, and such that the map ψU : π −1 (U ) −→ U × Cr is a biholomorphic bijection such that pr2 ◦ ψU = π. It follows that the transition functions gUV := ψU ◦ ψV−1 : ψV (V ∩ U ) −→ ψU (V ∩ U ) are biholomorphic. A meromorphic, (resp. holomorphic) section of a vector bundle is a meromorphic (resp. holomorphic) map s : X −→ V such that π ◦ σ = idX and such that sU : U −→ Cr defined as ψU ◦ s|U is a meromorphic (resp. holomorphic). Then, locally, a holomorphic section s is represented by r holomorphic functions on U : sU = (s1 , . . . , sr ). If U is an affine open set of X, the section is said to be regular if ⊕n (s1 , . . . , sr ) is a regular sequence in the ring OX (U ). Denote by OX r the trivial vector bundle pr1 : X × C −→ X of rank r, pr1 being the projection onto the 1st factor. Its sections can be seen as functions from X −→ Cr . The structural sheaf OX can be seen as the trivial vector bundle of rank 1 over X. If V1 and V2 are two vector bundles of rank r1 and r2 , respectively, a bundle morphism is a morphism φ : V1 −→ V2 such that π2 ◦ φ = π1 and φx : (V1 )x −→ (V2 )φ(x) is a vector space homomorphism. 2.3.14 Chern Classes. Any vector bundle over X of rank 1 will be said to be a line bundle. If s is a regular non-zero holomorphic section of a line bundle L, the zero scheme Z(s) ⊆ X is a codimension 1 subvariety said to be a Cartier divisor. By definition c1 (L) is the class in A1 (X) of Z(s). The set of all Cartier divisors is a group ∗ called P ic(X): analytically is the group H 1 (X, OX ) parametrizing isomorphism classes of line bundles over X. On a smooth variety, the natural map P ic(X) −→ A1 (X) is a group isomorphism (this is not true in general if X is not smooth). To each holomorphic vector bundle V −→ X of rank r, one can attach some distinguished classes: c1 (V ), . . . , cr (V ) ∈ A∗ (X),

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said to be the Chern classes of V . The rth Chern class has the following interpretation: if s : OX −→ V is a regular holomorphic section of V , then the cycle class associated to the zero scheme Z(s) of s is precisely cr (V ) ∩ [X]. The Chern polynomial is ct (V ) :=

X

ci (V )ti .

i≥0

where c0 (V ) = 1. The Chern polynomial is certainly a finite sum. In fact ck (V ) = 0 if k is bigger of the dimension of X, because Ak (X) = 0 whenever k > n. Moreover one can prove (see [16], p. 50) that ck (V ) = 0 if k exceeds the rank of the bundle. All the Chern classes ci of the trivial bundle pr1 : X × Cr −→ X are zero for i > 0. If 0 −→ V ′ −→ V −→ V ′′ −→ 0 is an exact sequence of vector bundles, one has ct (V ′ )·ct (V ′′ ) = ct (V ), the equality holding in the polynomial ring A∗ (X)[t].

2.3.15 Example. Let V be a vector space of dimension n and let V ∨ be its dual. The tautological (or universal) exact sequence of vector bundles on P(V ) is: 0 −→ T −→ OP(V ) ⊗ V −→ Q −→ 0, where OP(V ) ⊗ V is the trivial vector bundle P(V ) × V , T is the bundle T = {([u], v) ∈ P(V ) × V | v ∈ [u]}, whose fiber over [u] ∈ P(V ) is [u] := C · u itself, thought of as a 1dimensional subspace of V , and Q is the universal quotient bundle, whose fiber over [u] ∈ P(V ) is the quotient V /[u]. Let E1+n := (e0 , e1 , . . . , en ) be a basis of V , and let E1+n := (ǫ0 , ǫ1 , . . . , ǫn ) be the basis of V ∨ , dual of E1+n . Any α ∈ V ∨ induces a bundle map  α : T −→ P(V ) × C ([v], u) 7−→ ([v], α(u)) so that V ∨ can be seen as a global holomorphic section of the dual bundle T ∨ , usually denoted by OP(V ) (1), and said to be the hyperplane bundle or also the twisting sheaf of Serre (Cf. [34], p. 117). Conversely, any bundle map α∨ : T −→ P(V ) × C

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induces a homomorphism α : V −→ C. In fact if 0 6= β ∈ V ∨ , the set Uβ = {v ∈ V | β(v) 6= 0} is an open set of V . Let Uβ′ be the image of Uβ w via the map w 7→ β(w) . Then α∨ is represented by a family of holomorphic functions αβ : Uβ′ −→ C such that βαβ |Uβ′ ∩Uγ′ = γαγ |

′ U ′ ∩Uγ β

(one used the hypothesis that α∨ is a bundle morphism). In particular αβ admits a Taylor series expansion around any point of Uβ′ converging at αβ at each point of Uβ′ (which will turn out to be, see below, a polynomial function of degree 1). Let w ∈ V be arbitrarily chosen and let [v] be any subspace containing it (if w = 0 any [v] does the job). If v ∈ Uβ , then v w = t β(v) for some t ∈ C (possibly 0). Define α(w) = β(v)αβ

„

t

v β(v)

«

= t · β(v)αβ

„

v β(v)

«

∈ C.

Then α is a well defined holomorphic map V −→ C (i.e., if w ∈ Uβ ∩ Uγ , the result does not depend on the representation of α). Moreover, since by construction α(λw) = λα(w), Euler’s theorem on homogeneous functions says that α is linear. We have hence proven that H 0 (OP(V ) (1)), the space of the global holomorphic sections of OP(V ) (1) is isomorphic to V ∨ . If 0 6= α ∈ H 0 (OP(V ) (1)), then Z(α) = Hα := P(ker(α)), is a projective hyperplane. Since for any α, β ∈ H 0 (OP(V ) (1)) the ratio α/β defines a rational function on P(V ), any two hyperplanes Hα and Hβ determine the same class in An−1 (X) modulo rational equivalence. The 1st Chern class c1 (OP(V ) (1)) = h ∈ A1 (X) is defined as: h ∩ [P(V )] = [H] ∈ An−1 (P(V )), where [H] is the hyperplane class. Hence capping with the 1st Chern class corresponds to cutting with a hyperplane (modulo rational equivalence). 2.3.16 Example. Example 2.3.15 above can be generalized. On the grassmannian variety Gk (V ), parametrizing k dimensional vector subspaces of an n-dimensional vector space V , the tautological exact sequence is: 0 −→ Tk −→ OGk (V ) ⊗ V −→ Qk −→ 0.

(2.9)

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where the tautological bundle Tk is: Tk := {([Λ], v) ∈ Gk (V ) × V | v ∈ [Λ]}. Any element of V ∨ can be seen as an element of Tk∨ := Hom(Tk , OGk (V ) ), by defining: α([Λ], v) = ([Λ], α(v)). The quotient Qk is a bundle of rank n − k whose fiber over [Λ] ∈ Gk (V ) is V /[Λ]. 2.3.17 Example. The intersection ring of the complex projective space Pn is: Z[h] A∗ (Pn ) := 1+n , (h ) where h∩[P(V )] is the class [H] ∈ An−1 (X) of any hyperplane. One already knows (Cf. Example 2.3.12) that Ai (X) is generated by hi . Moreover hi · hj = hi+j , because one may find a (n − i)-dimensional plane H (i) and (n − j)-dimensional plane H (j) intersecting tranversally, and hence along a n − (i + j)-dimensional linear variety, representing the class hi+j . Since A0 (Pn ) is cyclic, it has no torsion (because any rational function on a curve has as many zeros as poles) and therefore Ai (Pn ) is not torsion as well, and hence isomorphic to Z · hi . In fact, were ahi ∈ Ai (Pn ) such that ahi ∩ [Pn ] = 0 in An−i (Pn ), then ahi hn−i ∩ [Pn ] = ahn ∩ [Pn ], i.e. a = 0. If Y ⊆ Pn is any projective subvariety of dimension i, its degree is defined as deg(Y )[pt] = hi ∩ [Y ], where [pt] := hn ∩ [X] is the class of the point. This corresponds with the intuitive geometric idea that the degree of Y can be gotten by intersecting it with sufficiently many hyperplanes in sufficiently general position. The relation h1+n = 0 says that (1 + n) hyperplanes in general position do not intersect at all. Therefore, by the discussion above, the Chow ring of P(V ) is generated by the first Chern class of the hyperplane bundle OP(V ) (1).

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Chapter 3

Frames and Grassmannians The purpose of this chapter is to provide a smooth introduction to grassmannians for beginners. Many important topics have been omitted (one for all, the discussion of the tangent space to a grassmannian at a k-plane, see e.g. [32] or also [1]), but what is in here is more than needed to follow the rest of the exposition. At the beginning we work over an arbitrary field F, to stress that most of the features of a grassmannian are of linear algebraic nature. We shall finally switch to the complex field when geometry will ask for it.

3.1

Warming Up

3.1.1 The Set of k-Frames. Let V be an F-vector space. If Λ := (u1 , . . . , uk ) ∈ V k is any ordered k-tuple of V , one agrees to denote by [Λ] := [u1 , . . . , uk ] the vector subspace Fu1 + . . . + Fuk of V generated by Λ. If dimF [Λ] = h, one briefly says that [Λ] is a h-plane of V . Any 38

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k-tuple Λ ∈ V k can be seen as a linear map: Λ : Fk −→ V v 7−→ Λ · v, where:



 v1   Λ · v = Λ ·  ...  = v 1 u1 + . . . + v k uk , vk

and therefore every v ∈ [Λ] is of the form Λ · u for some u ∈ Fk .

3.1.2 Definition. A k-tuple Λ := (v1 , . . . , vk ) ∈ V k is said to be a k-frame of V if [Λ] is a k-plane, i.e. if (v1 , . . . , vk ) are linearly independent. Let: Fk (V ) := {Λ ∈ V k | dimF [Λ] = k},

be the set of all k-frames of V . If Λ ∈ Fk (V ), each u ∈ [Λ] is of the form u = Λ · v for a unique v ∈ Fk . 3.1.3 Example. i) [0V ] = {0V }; ii) a 1-frame of V is just a non zero vector of V . Hence F1 (V ) = V ∗ , where V ∗ := V \ {0V }; iii) the set Fk (Fn ) may be identified with the set of all n × k matrices with linearly independent columns, i.e. matrices of rank k; iv) if dim V = n and Λ ∈ Fn (V ), then [Λ] = [V ].

3.1.4 Let V ∨ be the dual vector space HomF (V, F) of V , and suppose that dimF (V ) = n. Let us fix once and for all a basis En := (e1 , . . . , en ), of V and the basis En := (ǫ1 , . . . , ǫn )

of V ∨ dual of En (i.e. ǫi (ej ) = δji ). Each v ∈ V can be uniquely written as: n X v= v i ei , i=1

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CHAPTER 3. FRAMES AND GRASSMANNIANS

where v i = ǫi (v). If Λ = (u1 , . . . , uk ) ∈ V k , one sets: ǫi (Λ) := ǫi (u1 , . . . , uk ) = (ui1 , . . . , uik ) := (ǫi (u1 ), . . . , ǫi (uk )). One also writes:  1  1  1 ǫ (u1 ) ǫ ǫ  ..   ..  ..   .  (Λ) =  .  (u1 , . . . , uk ) =  . ǫn

 . . . ǫ1 (uk ) ..  , (3.1) .. . .  n n ǫ (u1 ) . . . ǫ (uk )

ǫn

so that the basis En gives an identification:  1 ǫ  ..   .  : V k −→ Mn×k (F), ǫn

where Mn×k (F) is the F-vector space of all the n × k matrices with Vk entries in F. Each ǫi1 ∧ . . . ∧ ǫik ∈ V (Section 2.2) induces a function: ǫi1 ∧ . . . ∧ ǫik : V k −→ F defined by:  ǫi1 (Λ)   ǫi1 ∧ . . . ∧ ǫik (Λ) := ǫi1 ∧ . . . ∧ ǫik (u1 , . . . , uk ) = det  ...  = 

Notice that:

1 ǫ (u1 ) = ... ǫn (u1 )

˛ i ˛δ 1 ˛ j1 ˛ . ǫi1 ∧ . . . ∧ ǫik (ej1 , . . . , ejk ) = ˛˛ .. ˛ ik ˛δj 1

In particular

. . . ǫ1 (uk ) u11 .. = .. .. . . . n . . . ǫ (uk ) un1 ... .. . ...

ǫik (Λ) . . . u1k .. . .. . . . . . unk

˛ δji1k ˛˛ X .. ˛˛ (−1)|τ | δji1τ (1) · . . . · δjikτ (k) . . ˛= ˛ τ ∈Sk i δjkk ˛

ǫi1 ∧ . . . ∧ ǫik (ei1 , . . . , eik ) = 1,

(3.2)

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whenever the indices ij are pairwise distincts and ǫi1 ∧ . . . ∧ ǫik (ej1 , . . . , ejk ) = 0,

(3.3)

whenever there is at least a jp ∈ / {i1 , . . . , ik } (1 ≤ p ≤ k). We have hence got a map Vk Vk −→ V (u1 , . . . , uk ) 7−→ u1 ∧ . . . ∧ uk

Vk Using the basis (ei1 ∧ . . . ∧ eik ) of V one has: X ∧k Λ := u1 ∧ . . . ∧ uk = ǫi1 ∧ . . . ∧ ǫik (Λ)ei1 ∧ . . . ∧ eik . (3.4) 1≤ij ≤n

Because of formula (2.3), the sum occurring on the r.h.s. of (3.4) is only over k-schindices (Cf. Sect. 2.1.5) such that ik ≤ n. 3.1.5 Proposition. A k-tuple Λ := (u1 , . . . , uk ) ∈ V k is a k-frame if and only if u1 ∧ . . . ∧ uk 6= 0 (3.5) as a vector of ∧k V . Vk Proof. Indeed u1 ∧ . . . ∧ uk is a vector of V whose components, V k with respect to the basis ∧k En of V , are precisely the k ×k minors of the matrix of formula (3.1). The latter has rank k if and only if there is at least one non vanishing minor among them. One must check, however, that condition (3.5) does not depend on the choice of a basis of V . But were (f1 , . . . , fn ) another basis of V , then fi = ej Aji for some invertible matrix A := (Aij ) and ǫj = Aji φi . Thus: v1 ∧ . . . ∧ vk = ǫi1 ∧ . . . ∧ ǫik (Λ)ei1 ∧ . . . ∧ eik =

Aij11 · . . . · Aijkk · φj1 ∧ . . . ∧ φjk (Λ)ei1 ∧ . . . ∧ eik = φj1 ∧ . . . ∧ φjk (Λ)Aij11 ei1 ∧ . . . ∧ Aijkk eik =

φj1 ∧ . . . ∧ φjk (Λ) · fj1 ∧ . . . ∧ fjk ,

proving that v1 ∧ . . . ∧ vk 6= 0 is an intrinsic condition independent from the choice of a basis of V .

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3.1.6 Corollary. Let Λu := (u1 , . . . , uk ) and Λv := (v1 , . . . , vk ) be two k-frames of V . They span the same k-plane of V if and only if v1 ∧ . . . ∧ vk = a · u1 ∧ . . . ∧ uk ,

(3.6)

holds for some non zero a ∈ F. Proof. Suppose that [Λu ] = [Λv ]. Then Λu and Λv are both bases of the same k-plane. Therefore1 : vi = Aji uj , for some invertible k × k matrix A = (Aij ). Then: v1 ∧ . . . ∧ vk = det(A) · u1 ∧ . . . ∧ uk . Conversely, suppose that (3.6) holds for some non-zero a ∈ F. Then we claim that [v1 , . . . , vk , ui ] is not a (k + 1)-plane for each 1 ≤ i ≤ k. In fact v1 ∧ . . . ∧ vk ∧ ui = a · u1 ∧ . . . ∧ uk ∧ ui = 0, because two equal indices occur on the r.h.s. As a consequence a k-plane is uniquely determined by the data (ǫi1 ∧ . . . ∧ ǫik (Λ)) up to a scalar multiple. 3.1.7 Towards Grassmannians. Grassmannians are among the first examples of parameter spaces contructed with the purpose to solve concrete enumerative problems (see e.g. [79]). The simplest one such is surely how many lines do pass through two distinct points in the projective R-plane? This is an easy problem: everybody knows that the answer is 1 and can be gotten by writing down the equation of a general line and imposing the passing through two distinct points. One so gets a linear homogeneous system of two equations in three unknowns (the coefficients of the equation of the line) having a 1dimensional subspace of solutions. Since a line is determined by the coefficients of its equation up to a non-zero scalar multiple, one sees that there is only one such a line. However it is worth to present here a way to solve the same problem that can be generalized to less easy cases. 1 We

use Einstein’s convention on sums.

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Picking two distinct points of P2 is the same as picking two distinct 1-planes [u1 ] and [u2 ] in R3 , which is the same as picking a 2-plane [v, w] ∈ G2 (R3 ). The problem is then to find how many 2-planes contain two distinct 1-lines. But ui ∈ [v, w] iff [v, w, ui ] 6= R3 , i.e. if and only if: v ∧ w ∧ u1 = 0 . v ∧ w ∧ u2 = 0 The reader may easily check that such a system can be written in the form:   1 ǫ (u1 ) ǫ1 (u2 ) (ǫ2 ∧ ǫ3 (v, w), ǫ3 ∧ ǫ1 (v, w), ǫ1 ∧ ǫ2 (v, w)) · ǫ2 (u1 ) ǫ2 (u2 ) = 0. ǫ3 (u1 ) ǫ3 (u2 ) Since the rank of the matrix (notation as in formula (3.1))  1 ǫ ǫ2  (u1 , u2 ) ǫ3

is 2, because u1 ∧ u2 6= 0, one finds a unique 1-dimensional space of solutions, i.e. a unique 2-plane containing u1 and u2 . 3.1.8 Another exercise. In the same vein, let us try to look for the number of lines of the projective space P3 (F), F = R or C, meeting 4 others in general position (i.e. any two of them do not lie in a same 2plane). This is the same as looking for the 2-planes in F4 intersecting four others in general position along a positive dimensional subspace. Let [Λi ] ∈ G2 (F4 ) (1 ≤ i ≤ 4) be given. Then Λ ∈ G2 (F4 ) intersects Λi in a positive dimensional subspace if and only if [Λ, Λi ] 6= F4 . This gives the following system (notation as in (3.1)): 1 0 1 ǫ ∧ ǫ2 0 1 Bǫ1 ∧ ǫ3 C 0 B 1 C Bǫ ∧ ǫ4 C B0C 34 24 23 14 13 12 C B C B (X , X , X , X , X , X )(Λ) · B 2 3 C · (Λ1 , Λ2 , Λ3 , Λ4 ) = @ A . 0 Bǫ2 ∧ ǫ4 C @ǫ ∧ ǫ A 0 ǫ3 ∧ ǫ4 (3.7)

where (X 34 , X 24 , X 23 , X 14 , X 13 , X 12 )(Λ) = = (X 34 (Λ), X 24 (Λ), X 23 (Λ), X 14 (Λ), X 13 (Λ), X 12 (Λ))

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and X ij (Λ) = ǫi ∧ ǫj (Λ). Notice that equation (3.7) can be seen as the intersection of four hyperplanes in P5 (F), with homogeneous coordinates (X ij ). Since the system has rank 4, this is precisely the equation of a line in such a P5 . Last remark consists in observing that not any point of such a P5 can be seen as the set of coordinates of some 2-plane. In fact, the check being left to the Reader’s care, any [Λ] ∈ G2 (F4 ) satisfies the following quadratic equation: X 12 X 34 − X 13 X 24 + X 14 X 23 = 0.

(3.8)

Therefore, the solution of our enumerative problem can be seen as the number of intersection points between a line and the quadric hypersurface (3.8) P5 (F), called the Klein’s quadric. If F = R one may have either 0 or 2 solutions, since R is not algebraically closed. If F = C, which is algebraically closed, instead, one would always find 2 solutions (according to the multiplicity). For sake of uniformity, then, the field R will be from now on ignored and it will be assumed that the ground field F is C. The letter F, used to denote a general field, exits now from our play.

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Chapter 4

Frames and Grassmannians 4.1

Complex Grasmannian Varieties

The aim of this section is to imitate the construction of projective spaces to put a structure of complex manifold on the Grassmann Set of k-planes of a finite dimensional complex vector space. This will give us the grassmannian variety or, briefly, the Grassmannian. First of all, let us mention the following remarkable property. 4.1.1 Proposition. The group Gl(V ) acts transitively on Fk (V ) and Gk (V ). Proof. Consider the map: Gl(V ) × Fk (V ) −→ Fk (V ), defined by (A, (u1 , . . . , uk )) 7→ (A · u1 , . . . , A · uk ). It is clearly an action on Fk (V ), inducing an action on Gk (V ) as well, by setting: A · [Λ] = [A · Λ]. Let now Λu := (u1 , . . . , uk ) and Λv := (v1 , . . . , vk ) be any two kframes. Let W be a finite dimensional subspace of V containing both 45

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[u1 , . . . , uk ] and [v1 , . . . , vk ]. If n := dim(W ), let: (u1 , . . . , uk , uk+1 , . . . , un )

and (v1 , . . . , vk , vk+1 , . . . , vn ),

be any two bases of W gotten by completing Λu and Λv . Let φ ∈ Gl(W ) be the unique automorphism of W sending ui onto vi . If V = W ⊕ W ′ , where W ′ is the linear complement of W in V , denote by idW ′ the identity of W ′ . Then the the automorphism φ ⊕ idW ′ : V −→ V, maps Λu onto Λv , as well as [Λu ] onto [Λv ]. This proves that Gl(V ) acts transitively on Fk (V ) and Gk (V ). 4.1.2 According to Proposition 3.1.5, Fk (V ) is the complement in V k of the zero locus of finitely many polynomial equations (ǫi1 ∧ . . . ∧ ǫik (Λ) = 0, ∀(i1 , . . . , ik ) ∈ Ink ). These are continuous in both the Zariski or the usual topology. Then Fk (V ) is a (Zariski-)open set in V k , because (v1 , . . . , vk ) is a k-frame if and only if v1 ∧ . . . ∧ vk 6= 0 (i.e. at least one component of it, with respect to the basis of ∧k V , is different from zero). However, as already implicitly remarked, the same k-plane can be determined by several different k-frames. The reason is that there is a natural right action of the group Glk (C) of the k × k C-valued invertible matrices on the set of k-frames, given by: Fk (V ) × Glk (C) −→ Fk (V ) , (Λ , A) 7−→ Λ · A where if Λ = (v1 , . . . , vk ) and A = (Aij ) then Λ · A = (vi Ai1 , . . . , vi Aik ). Straightforward computations show that ǫi1 ∧ . . . ∧ ǫik (Λ · A) = ǫi1 ∧ . . . ∧ ǫik (Λ) · det(A). Given that any two k-frames Λ, Λ′ defines the same k-plane if there is A ∈ Glk (C) such that Λ′ = Λ · A, it follows that Gk (V ) = Fk (V )/Glk (V ). Moreover Λ and Λ′ live in the same Fk (V )-orbit

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if the dim[Λ, Λ′ ] = k. This amounts to impose that the rank of any matrix associated to the subspace [Λ, Λ′ ] is smaller or equal than k, i.e. all its (k + 1) × (k + 1) minors must vanish. Therefore the graph in Fk (V ) × Fk (V ) of the relation Λ ∼ Λ′ if [Λ] = [Λ′ ] is (Zariski) closed, showing that the quotient topology of Gk (V ) is Hausdorff (or separated in the Zariski topology). Let ρ : Fk (V ) −→ Gk (V ), be the canonical projection and write ρ(Λ) = [Λ]. The fiber over [Λ] ∈ Gk (V ) is the set ρ−1 ([Λ]) = Λ · Glk (C). For each k-schindex I = (1 ≤ i1 < . . . < ik ≤ n), let UI = {[Λ] ∈ Gk (V ) | ǫi1 ∧ . . . ∧ ǫik (Λ) 6= 0}. As remarked, the condition ǫi1 ∧ . . . ∧ ǫik (Λ) 6= 0 depends only on the k-plane [Λ] and not on its particular representative. 4.1.3 Proposition. For each k-schindex I = (i1 , . . . , ik ), the set UI is open in the quotient topology of Gk (V ). Proof. Let Fk (V )I be the set of all Λ ∈ Fk (V ) such that    i  1 0 ... 0 1 ǫ 0 1 . . . 0    ..   .  (Λ) =  .. .. . . ..  = 1Glk (C) .  . . . . ǫik 0 0 ... 1 One has ρ(Fk (V )I ) = UI . Therefore

ρ−1 (UI ) = Fk (V )I · Glk (V ) which is the Zarisky open set of Fk (V ) complement of Z(ǫi1 ∧ . . . ∧ ǫik ) = {Λ ∈ Fk (V ) | ǫi1 ∧ . . . ∧ ǫik (Λ) = 0}. It follows that the collection UE := {UI |I ∈ I k , ik ≤ n} forms an affine open covering of Gk (V ) and each UI is isomorphic to Ck(n−k) . Moreover, a subset U ⊆ Gk (V ) is open if and only if U ∩ UI is (Zarisky) open in Ui1 ...ik . In fact if U ∩ UI is (Zariski) open in UI , then ρ−1 (U ) ∩ ρ−1 (UI ) is (Zariski) open in ρ−1 (UI ). But ρ−1 (UI ) is

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an open covering of Fk (V ) and, therefore, ρ−1 (U ) is open, i.e. U is open in Gk (V ). Let ψI : UI −→ Fk (V )I be the map defined by 

−1 ǫi1 (Λ)   ψI ([Λ]) = ΛI := Λ ·  ...  . ǫik (Λ)

Then, if [Λ] ∈ UI ∩ UJ , one has:

ΛJ = ΛI · gIJ , where gIJ ∈ Glk (C) is given by:  i1   j1 −1 ǫ (Λ) ǫ (Λ)     gIJ =  ...  ·  ...  . ǫik (Λ)

ǫjk (Λ)

It is straightforward to check that gIJ · gJK = gIK . This cocycle relation shows two things: first, that one can see the grassmann set Gk (V ) as an algebraic variety gotten by glueing nk affine spaces (the UI ’s) via the linear transition function(s) gIJ , which are polynomials (being product of matrices in the components of the frame Λ with respect to the basis En ). Secondly, the maps gIJ : UI ∩UJ −→ Glk (C) are matrices whose entries are quotient of polynomials with non zero denominator. Then such functions exhibit Fk (V ) as an algebraic principal GLk (C)-bundle over Gk (V ).

4.1.4 Theorem. The Grassmannian Gk (V ) is a smooth irreducible algebraic variety. Proof. Since Fk (V ) is an open set of V k , which is irreducible in the Zariski topology, then it is irreducible. The irreducibility of Gk (V ) then follows from the the fact that Gk (V ) is the continuous image of Fk (V ) via the projection map ρ. It is smooth because it has certainly smooth points, being covered by affine sets (the UI ’s) which are smooth. If it has a smooth points than all the points are smooth because it is a homogeneous variety (the group Gl(V ) acts transitively as an algebraic group of isomorphisms, see Proposition 4.1.1).

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k ^ lE : Fk (V ) −→ F1 ( V ),

Vk be defined by (v1 , . . . , vk ) 7→ v1 ∧ . . . ∧ vk ; here F1 ( V ) stands for Vk Vk V \ {0}, the set of 1-frames of V . Clearly F1 (∧k V ) is acted on by the multiplicative group Gm (C) := Gl1 (C) := C∗ . The map lE is equivariant, in the sense that lE (Λ · A) = lE (Λ) · det(A). Hence lE factors through a map k k ^ ^ P lE : Gk (V ) −→ P( V ) := G1 ( V ).

said to be the Pl¨ ucker map. In other words, two k-frames span the same plane if their Pl¨ ucker images are the same. Equip Glk (V ) with the quotient topology of Fk (V )/Glk (V ). 4.1.6 Proposition. The Pl¨ ucker map P lE is an embedding. Proof. By Corollary 3.1.6, the Pl¨ ucker map is injective. We are left to show that it has injective tangent map. Let Bε := {z ∈ C | |z| < ε}, be a disc in the complex plane. Then, any tangent vector to [Λ] ∈ Gk (V ) is of the form dγ , dz z=0

where γ : Bǫ −→ Gk (V ) such that γ(0) := [Λ]. Therefore the tangent of the Pl¨ ucker map is: dγ d i1 ik T[Λ] P lE = (ǫ ∧ . . . ∧ ǫ (z)) . dz z=0 dz z=0 (i1 ,...,ik )∈I k n

It is then sufficient to show that dγ d 6= 0 ⇐ T[Λ] P lE γ(z) 6= 0. dz z=0 dz z=0

The property being local, it suffices to check it on an affine open set of the Grassmannian of the form UI containing [Λ] . Up to a

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linear transformation permuting the elements of the basis En , one may assume that I = (12 . . . k). Any k-plane in UI can be represented by a maximal rank matrix of the form:   1 0 ... 0 0  0 1 ... 0 0     .. .. . ..  . .. ..  .  . .    0  0 . . . 1 0    0 , 0 . . . 0 1    x1,1  x . . . x x 1,2 1,k−1 1,k    x2,1 x2,2 ... x2,k−1 x2,k     . .. .. ..  ..  .. . . . .  xn−k,1

xn−k,2

. . . xn−k,k−1

xn−k,k

where a tangent vector can be represented in the form (dxij /dz)|z=0 . Since for each pair (i, j), such that 1 ≤ i ≤ k and k + 1 ≤ j ≤ n, (ǫ1 ∧ . . . ∧ ǫbj ∧ . . . ∧ ǫk ∧ ǫk+i )(Λ) = (−1)k−j+i−1 xij ,

the tangent map can be written as d i1 ik k−j+i−1 dxij , (−1) , ǫ ∧ . . . ∧ ǫ (γ(z)) dz z=0 dz z=0 (i1 ,...,ik )∈B

where B is the set of all k-schindices such that ♯(B ∩ {1, 2, . . . , k}) ≤ k − 2. Then the tangent map to the Pl¨ ucker embedding is injective as claimed (the null tangent vector is the unique pre-image of null vector through the tangent map).

4.2

Schubert Varieties

4.2.1 A complete flag of a n-dimensional vector space is a filtration E • of V : E• :

E 0 = V ⊃ E 1 ⊃ E 2 ⊃ . . . ⊃ E n = (0),

where E i is a vector subspace of V of codimension i. Suppose that En := (e1 , . . . , en ) is an adapted basis to E in the sense that E i = [ei+1 , . . . , en ], 1 ≤ i ≤ n − 1.

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4.2.2 Definition (Cf. Example 6.4.3). The ith E-Schubert matrix of [Λ] ∈ Gk (V ) is the i × k-matrix:  1  ǫ (Λ)   Sch([Λ] ∩ E i ) :=  ...  . ǫi (Λ)

.

Let ρi (E, Λ) := rk(Sch([Λ] ∩ E i )). Then one has: 4.2.3 Proposition. The following equality holds: dim(E i ∩ [Λ]) := k − ρi (E, Λ). Proof. In fact, the vectors v ∈ [Λ] belonging to E i must satisfy the linear system of equations:  1  ǫ (v) = 0 .. . .   i ǫ (v) = 0

The dimension of the space of solutions is precisely k minus the rank of the system, which is exactly ρi (E, Λ). Because of the obvious inequalities: ρi (E, Λ) ≤ ρi+1 (E, Λ) ≤ ρi (E, Λ) + 1 (adding a row to a matrix the rank increases at most of 1), one deduces that dim(E i+1 ∩ [Λ]) ≤ dim(E i ∩ [Λ]) ≤ dim(E i+1 ∩ [Λ]) + 1, i.e., any possible dimension “jump” is not bigger than 1. The upshot is that in the sequence: k := dim([Λ] ∩ E 0 ) ≥ dim([Λ] ∩ E 1 ) ≥ . . . ≥ dim([Λ] ∩ E n ) = 0, there are exactly k dimension jumps. 4.2.4 Definition. An integer 1 ≤ i ≤ n is a Schubert jump if dim([Λ] ∩ E i−1 ) > dim([Λ] ∩ E i ).

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4.2.5 Definition. The Schubert index of the k-plane [Λ] is the k-schindex I([Λ], E • ) = (1 ≤ i1 < . . . < ik ≤ n) of the Schubert jumps at [Λ]. To any k-plane one can obviously attach one and only one Schubert index. The general k-plane has Schubert index (1, 2, . . . , k). In fact, a k-plane is general with respect to the flag E • if it intersects E k (the subspace of codimension k) in the null vector, because the general homogeneous linear system of k equations in k unknowns has no solution but the trivial one. As a matter of fact, the E • -general k-plane lives in the complement of a Zariski closed set. Let us see that. If [Λ] is not general, then dim([Λ] ∩ E k ) > 0. Hence there exists 0 6= v ∈ [Λ] ∩ E k , i.e. there exists u ∈ Ck \ {0}, such that Λ · u satisfies the linear system: ǫi (Λ) · u = 0,

1 ≤ i ≤ k.

This is possible if det(ǫi (Λ)) = 0, i.e if ǫ1 ∧ . . . ∧ ǫk (Λ) = 0. If Λ = (v1 , . . . , vk ), the condition can be recast as: 1 ǫ (v1 ) . . . ǫ1 (vk ) .. = 0. .. ǫ1 ∧ . . . ∧ ǫk ([Λ]) = ... . . ǫk (v1 ) . . . ǫk (vk )

4.2.6 Definition. The expression ǫ1 ∧ . . . ∧ ǫk ([Λ]) is said to be the E • -Schubert Wronskian at [Λ].

Hence the k-planes in E • -special position live in the zero-scheme Z(ǫ1 ∧ . . . ∧ ǫk ) of ǫ1 ∧ . . . ∧ ǫk . It is worth to remark that the latter depends only on the flag E • and not on the adapted basis to the flag itself. For, were (φ1 , . . . , φn ) another basis such that E i = [φi+1 , . . . , φn ], then the transformation matrix T from (ǫj ) to the (φj ) would be triangular, so that φ1 ∧ φ2 ∧ . . . ∧ φk = det(T ) · ǫ1 ∧ . . . ∧ ǫk . Since (ǫ1 , . . . , ǫn ) are sections of T ∨ (see Example 2.3.16), it follows that ǫ1 ∧ . . . ∧ ǫk is a section of the line bundle ∧k T ∨ and thus that

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the class of the E • -special k-planes in the Chow group A∗ (Gk (V ) is precisely: [Z(ǫ1 ∧ . . . ∧ ǫk )] = c1 (∧∨ T ∨ ) ∩ [Gk (V )]. One easily sees that this class does not depend on the flag chosen: any two Schubert-wronskians are sections of the same line bundle. Vk Set σ1 = c1 ( Tk∨ ). Because of the tautological exact 2.9 and the fact that ct (Tk )ct (Qk ) = 1, it also follows that: k k ^ ^ σ1 = c1 ( Tk∨ ) = −c1 ( Tk ) = −c1 (Tk ) = c1 (Qk ).

4.2.7 Example. Let V := C[X]/(X n ) be the C-vector space of polynomials of degree at most n − 1. It is a n-dimensional C-vector space spanned by the classes of 1, X, X 2 , . . . , X n ∈ C[X] modulo (X n ). Let z0 ∈ C and consider the flag: V ⊃ V (−z0 ) ⊃ V (−2z0 ) ⊃ . . . ⊃ V (−nz0 ) = 0 where

(X − z0 )i + (X n ) Xn is the vector subspace of polynomials of degree ≤ n contained in the ith power of the maximal ideal (X − z0 ) of C[X], or, spelled in a more friendly way, the subspace of polynomials of degree ≤ n vanishing at z0 with multiplicity at least i (or divisible by (X − z0 )i ). Let [Λ] be a subspace of dimension k of V and let (p1 (X), . . . , pk (X)) be a basis of it. Then [Λ] is special with respect to the given flag, if there exists 0 6= p(X) ∈ [Λ] vanishing at z0 with multiplicity at least k. Write: V (−iz0 ) :=

p(X) = a1 p1 (X) + a2 p2 (X) + . . . + ak pk (X). Then p(i) (z0 ) = 0 for all 1 ≤ i ≤ k, where p(i) (X) is the ith derivative of the polynomial p(X). Hence [Λ] is special if the following determinant: ˛ ˛ p1 (z0 ) ˛ ′ ˛ p1 (z0 ) ˛ W (p1 , . . . , pk )(z0 ) := ˛ .. ˛ . ˛ ˛p(k−1) (z ) 0 1

... ... .. . ...

˛ pk (z0 ) ˛ ˛ ′ pk (z0 ) ˛ ˛ .. ˛ = 0. ˛ . ˛ (k−1) pk (z0 )˛

This is a true wronskian and hence motivates the terminology of Definition 4.2.6.

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CHAPTER 4. FRAMES AND GRASSMANNIANS Let Wi1 ...ik (E • ) := {[Λ] ∈ Gk (V )|I([Λ], E • ) = (i1 , . . . , ik )}.

4.2.8 Proposition. The set Wi1 ...ik (E • ) is an affine cell of codimension (i1 − 1) + . . . + (ik − k).

Proof. If [Λ] ∈ Wi1 ...ik then, in particular, [Λ] ∈ UI (Sect. 4.1.2). Therefore Wi1 ...ik is entirely contained in UI , which is affine of dimension k(n − k). If [Λ] ∈ Wi1 ...ik , then [Λ] = [ΛI ], where ΛI = (v1 , . . . , vk ) is a k-frame whose matrix t (ǫ1 , . . . , ǫn ) · ΛI is given by:    1  0 0 0 ... 0 ǫ .  .. .. .. . .  ..  . . .  .  . ..     i1 −1   0 0 0 . . . 0  ǫ    i1  1 0 0 . . . 0   ǫ     i1 +1  ∗ 0 0 . . . 0   ǫ      .. .. .. . .  ..  ..  . . .  .  . .    i −1  ∗ 0 0 . . . 0  ǫ 2     i  0 1 0 . . . 0   ǫ2     i +1  ∗ ∗ 0 . . . 0  ǫ 2      . . .  ..  . (4.1)  .  (v1 , v2 , . . . , vk ) =  .. .. .. . . . ..      ∗ ∗ 0 . . . 0  ǫi3 −1      0 0 1 . . . 0   ǫi3      ∗ ∗ ∗ . . . 0  ǫi3 +1       . . . .  .  . . ...   .. .. ..  ..      ∗ ∗ ∗ . . . 0  ǫik −1      0 0 0 . . . 1   ǫik      ∗ ∗ ∗ . . . ∗  ǫik +1       . . . .  .  . . ...   .. .. ..  ..  ǫn

∗ ∗

∗ ... ∗

Then Wi1 ,...,ik is an irreducible affine subvariety of UI , isomorphic to AN , where N is the numbers of stars occurring in the huge matrix (4.1). The stars correspond to those components of the vectors (v1 , . . . , vk ) not constrained by the condition ΛI ∈ WI . The constraints in the first column are: the first i1 − 1 components are 0, th th while the ith 1 is 1. Moreover one must put a zero in the i2 , . . . , ik i1 ik because (ǫ , . . . , ǫ )(ΛI ) must be the identity matrix. Hence, in the

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first column one has n − (i1 − k + 1) stars. Similarly, in the j th column one has the first ij − 1 components equal to 0, the ith j equal to 1 th and then one still put k − j − 1 zero corresponding to the ith j+1 , . . . ik rows. It follows that in the ith j column one has n − (ij − (k − (j − 1))) stars. Then one has:

= =

n − i1 − (k − 1) + n − i2 − (k − 2) + . . . + n − ik − (k − k) = kn − nk − (i1 − 1) − (i2 − 2) + . . . + (ik − k) = k(n − k) − wt(I)

where wt(I) =

4.3

Pk

j=1 (ij

− j) is then precisely the codimension of WI .

Intersection Theory on Gk (V )

This section is just a quick review intended to let the Reader doing the necessaries comparisons with the material of Chapter 5. For proofs see [16], [27], [53]. For a geometrically stimulating exposition, see [39]. 4.3.1 Definition. Let ΩI (E • ) = Ωi1 ...ik (E • ) = Wi1 ...ik (E • ), the closure being taken in Gk (V ). It is called the Schubert variety associated to the flag E. 4.3.2 Suppose F • is another flag of V . Because of the transitive action of Gl(V ) on Gk (V ), there exists an automorphism g of V sending the flag F • onto the flag E • and, consequently, the Schubert variety ΩI (F • ) isomorphically onto ΩI (E • ). Since Gl(V ) is rational (and connected), their classes modulo rational equivalence in A∗ (Gk (V )) are equal (Cf. Section 2.3.9). Then one let: ΩI = [ΩI (E • )] ∈ A∗ (Gk (V )), for some complete flag E • of V . The class ΩI ∈ A∗ (Gk (V )) will be said to be the Schubert cycle associated to the k-schindex I. One

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may also denote the same Schubert cycle as Ωλ , where λ = λ(I) (Cf. Sect. 2.1.6). Clearly Ω12...k = Ω(0...0) = [Gk (V )], the fundamental class of Gk (V ). The class of ΩI (E • ) corresponds to a class in A∗ (Gk (V )) classically denoted by σλ , related to it via the equality: σλ ∩ [Gk (V )] = Ωλ , expressing Poincaré duality for grassmannians. The equality σλ ∩ Ωµ = σλ ∩ (σµ ∩ [Gk (V )]) = (σλ · σµ ) ∩ [Gk (V )]. expresses instead the fact that A∗ (Gk (V )) is a module over A∗ (Gk (V )). One also has: 4.3.3 Proposition (Chow basis theorem). The classes Ωλ := σλ ∩[Gk (V )], of Schubert varieties modulo rational equivalence, freely generate the Chow group A∗ (Gk (V )). Proof (idea of ). This is a particular case of a general result regarding cellular complexes. In our specific example, one may argue as follows. Any k-plane of V belongs to at least one open E • -Schubert cell, namely that indexed by its Schubert cycle. Moreover, two distinct open Schubert cells are disjont, because otherwise there would be k-planes having more than 1 Schubert index with respect to the same flag, which is absurd. Then, using exact sequence (2.8), one sees that the Chow group of Gk (V ) (or its integral homology), is generated by the classes modulo rational equivalence of the Schubert Varieties (the closure of the open Schubert cells). To show that the classes σλ ∩ [Gk (V )] freely generated the Chow group, one uses duality: for each λ ∈ ℓ(k), σλ · σλ∨ = σ((n−k)k ) = σ(n−k,...,n−k) and the latter freely generates Ak(n−k) (Gk (V )). For the proof of this fact see, e.g., [16], p. 268 or [27]. The following example serves as illustration of how Schubert’s devices should work. 4.3.4 Example. In the projective space P5 fix: a) a 3-dimensional projective linear subspace H and a point P ∈ H;

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b) a set of 2-codimensional projective linear subspaces Π1 , . . . , Π5 in general position. Suppose one wants to look for the class in A∗ (G(2, P6 )) of all the planes intersecting H along a line passing through P and incident to Πi , for 1 ≤ i ≤ 5. Equivalently, one is asking to look for the class in A∗ (G(3, 6)) of all the (affine) 3-dimensional vector subspaces of C6 such that a) intersect a 4-dimensional subspace H of C6 along a 2-plane containing a given 1-plane (corresponding to the point P ) and b) intersecting 5 subspaces of dimension 3 along a positive dimensional subspace. The first step to solve such an exercise is to identify the involved Schubert varieties . Let E • be the complete flag: C7 = E 0 ⊃ E 1 ⊃ E 2 ⊃ E 3 ⊃ E 4 ⊃ E 5 ⊃ E 6 = (0) If [Λ] ∈ G(3, 6) satisfy a), then one has that E 5 ⊂ [Λ] ∩ E 2 , and that dim([Λ] ∩ E 3 ) ≥ 2. The general such plane is contained in the Schubert cell W136 (E · ). In fact if [Λ] verifies a) one has dim(E 2 ∩ [Λ]) ≥ 2 and dim(E 5 ∩ [Λ]) = 1. The most general such plane is when the equality holds, which corresponds indeed to the k-planes having 1, 3, 5 as a Schubert index. If a 3-plane [Λ], instead, meet E 3 (which has (affine) dimension 3) along a positive dimensional vector subspace, then it belongs to the Schubert cycle Ω124 (E • ), since for the most general such 3-plane one has dim([Λ] ∩ E 4 ) = 0. By Kleiman’s theorem (Section 2.3.9), one knows that it is possible to choose sufficiently general flags F1• . . . , F5• , F0• such that the intersection: Y := Ω124 (F1• ) ∩ . . . ∩ Ω124 (F5• ) ∩ Ω136 (F0• ) is proper, i.e. such that the codimension of the intersection scheme coincides with the sum of the codimensions of those one is intersecting. The class of Y in A∗ (G(3, 7) is then: [Y ] = σ15 · σ31 ∩ [G(3, 6)] and the problem now amounts to compute explicitly the product σ15 σ31 ∈ A∗ (G(3, 6)) = A∗ (G(2, P5 ). Notice that Y has codimension 9 = dim(G(3, 6)). To continue computations see Example 4.3.9

4.3.5 Since any Schubert variety is characterized by the Schubert index of a general point of it, one sees, almost by definition, that ΩI (E • ) is the Zariski closure WI (E • ) of the Schubert cell •

WI (E ) =



˛ j ˛ ǫ 1 ∧ . . . ∧ ǫjk (Λ) = 0, [Λ] ∈ Gk (V ) ˛˛ ǫi1 ∧ . . . ∧ ǫik (Λ) 6= 0

∀(j1 , . . . , jk ) ≺ I

ff

(4.2)

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In consideration of this remark, i.e. that the Schubert varieties are defined, even scheme theoretically, by equations (4.2), one shall also use (see e.g. Example 6.1) the notation: [ǫi1 ∧ . . . ∧ ǫik ],

(4.3)

to denote the Schubert cycle Ωi1 ...ik ∈ A∗ (Gk (V )). So, imitating Lascoux (see [61], p. 12) who said “The Schubert variety is a Schur function”, we may say as well: The Schubert variety is a generalized wronskian. Indeed the two sentences are equivalent, as it is implicitly shown in the recent work [48]. However we shall not insist on that. 4.3.6 Example. Let us fix a flag E • in C4 . The homogeneous coordinate ring of the Pl¨ ucker embedding of the grassmannian is the homogeneous polynomial ring C[ǫi ∧ ǫj ] S(G(2, 4)) = , (Q) where Q = 0 is the Klein quadric relation. The ideal of the Schubert cycle [ǫ2 ∧ ǫ3 ] in S(G(2, 4)), represented by the Schubert variety Ω23 (E • ), is (ǫ1 ∧ ǫ2 , ǫ1 ∧ ǫ3 , ǫ1 ∧ ǫ4 ), while that of [ǫ1 ∧ ǫ4 ], represented by the Schubert variety Ω14 (E • ), is (ǫ1 ∧ ǫ2 , ǫ1 ∧ ǫ3 , ǫ2 ∧ ǫ3 ). The Reader should check, for sake of exercise, that the sum of the two ideals cut out precisely the Schubert variety [ǫ2 ∧ ǫ4 ].

4.3.7 Via the Poincaré isomorphism: A∗ (Gk (V )) −→ A∗ (Gk (V )), sending σλ 7→ σλ ∩ [Gk (V )], and by Proposition 4.3.3, it follows that A∗ (Gk (V )) is generated as a Z-module by the Schubert cycles σλ . It turns out that σi = ci (Qk ) (see [16], p. 271). Doing intersection theory on the grassmannian amounts to knowing how to multiply any two Schubert classes σλ and σµ , i.e. to know σλ · σµ in A∗ (Gk (V )) or, equivalently, σλ ∩Ωµ ∈ A∗ (Gk (V )). Using the combinatorial language of Young diagrams (see Section 2.1 or, better, [20]), one may also say that the Chow ring A∗ (Gk (V )) is freely generated, as a module over the integers, by the Schubert (co)cycles:

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{σλ | λ is a partition contained in a k(n − k) rectangle},

where σλ ∩ [Gk (V )] is the class of a Schubert variety Ωλ (E • ) associated to any flag E • of V . Schubert Calculus allows to write the product σλ · σµ as an explicit linear combination of elements of the given basis of A∗ (Gk (V )). It consists, indeed, in an explicit algorithm ν to determine the structure constants {Cλµ } defined by: X ν σλ · σµ = Cλµ σν . |ν|=|λ|+|µ|

ν Cλν

The coefficients can be determined combinatorially via the Littlewood-Richardson rule ([52], p. 68). The other recipe consists in determining any product via reduction to known cases. To this purpose, one first establishes a rule to multiply any Schubert cycle with a special one. A special Schubert cycle is a cycle indexed by a partition of length 1. Such a product is ruled by 4.3.8 Theorem (Pieri’s Formula). The following multiplication rule holds: X σh · σλ = σµ (4.4) µ

the sum over all partitions such that |µ| = |λ| + h and n − k ≥ µ1 ≥ λ1 ≥ . . . ≥ µk ≥ λk .

where n = dim(V ). Proof. (see e.g. [27], p. 203). Pieri’s formula can be also phrased by saying that sum (4.4) is over all the partitions µ whose Young diagram Y (µ) is gotten by adding h boxes to Y (λ), in all possible ways, not two on the same column. For instance, in G(3, n), with n ≥ 9, one has: σ2 · σ331 = σ531 + σ432 + σ333 .

The “graphical Pieri’s formula”, in terms of Young diagrams is depicted below: ∪

=

* *

*

+ *

+ * *.

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4.3.9 Example (4.3.4 continued). With Pieri’s formula at our disposal one may compute the intersection product σ15 · σ31 ∈ A∗ (G(3, 6)). One applies Pieri’s formula 5-times, getting: σ14 (σ32 + σ311 ) = . . . (f our more times) . . . = 5 · σ333 . Therefore the answer to the question proposed in Example 4.3.4 is 5 (see Example 6.2.3 for computational details in a different formalism).

It is not difficult to prove, and in fact we shall do it in Chapter 5, that Pieri’s formula determines, indeed, the ring structure of A∗ (G). In particular, one can see that A∗ (G) is generated, as a ring, by the first k special Schubert cycles σ1 , . . . , σk . This is a consequence of another explicit consequence of Pieri’s formula, i.e. the determinantal Giambelli’s formula, expressing the Schubert cycle σλ as a polynomial in the special ones: 4.3.10 Proposition (Giambelli’s Formula). The Schubert cycle associated to a partition λ = (rk , . . . , r1 ) is a (determinantal) polynomial expression in the special Schubert cycle σi ’s: ˛ ˛ σr 1 ˛ ˛ σr1 −1 ˛ σλ = ∆λ (σ) = ˛ .. ˛ . ˛ ˛σr −k+1 1

σr2 +1 σr 2 .. . σr2 −k+2

... ... .. . ...

˛ σrk +k−1 ˛˛ σrk +k−2 ˛˛ ˛ = det(σrj +j−i ). .. ˛ . ˛ σ rk ˛

Proof. It will be given in Section 5.8, within the formalism of Sderivations as a consequence of a suitable “integration by parts”. Therefore, the computation of an arbitrary product σλ · σµ is reduced to a sequence of applications of Giambelli’s and Pieri’s formula: one first writes σλ as a polynomial in the σi ’s, and then applies Pieri’s formula as many times as necessary and then again Giambelli’s and so on. Computations become intricate in big grassmannians and for lengthy partitions, but products are computable in principle. 4.3.11 Example. To compute σ11 · σ21 in A∗ (G(3, 6)), one first use Giambelli’s formula to write, e.g., σ11 as σ12 − σ2 and then expand the product: (σ12 − σ2 )σ21 , by applying thrice Pieri’s formula. The result, whose checking is left to the reader, is σ221 + σ311 + σ32 .

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4.3.12 A few comments. As already said, Giambelli’s formula is a formal consequence of Pieri’s formula. However, the way to show it may be rather tricky. The reader may look at the proof of [27], p. 204-206, which, in spite of being conceptually easy, is based on a formula whose proof consists in a case by case combinatorial analysis. Another equivalent proof is that presented in [53], p. 13. There. the Author proves Giambelli’s formula in the realm of symmetric functions, where a Schur polynomial associated to a given partition is expressed as a determinant of complete symmetric functions: that is the so-called Jacobi-Trudy formula (see [52], pp. 23 ff. and also [16], p. 422). The proof works by induction and the details of a preliminary lemma (analogous to that used in [27], p. 204) are left to the reader’s care. The proof we propose in Section 5.8, within the formalism of S-derivations, instead, is based on the definition of determinant of a square matrix! The shortest elegant proof the author knows is the very general one, by Laksov and Thorup, in a recent preprint ([48]), within the framework of symmetric functions. The philosophy is however quite different from that of these notes: there is there more mathematics, more generality and hence more naturality. 4.3.13 Giambelli’s formula proves that A∗ (Gk (V )) is generated, as a ring, by the special Schubert cycles only. Its presentation can be gotten combining Giambelli’s and Pieri’s formula and is given by A∗ (Gk (V )) =

Z[σ1 , . . . , σk ] , (Σn−k+1 (σ), . . . , Σn (σ))

where each Σn−k+i (σ) is an explicit weighted homogeneous polynomial in σ1 , . . . , σk of degree n − k + i. Such a presentation shall be explicitly computed in Section 5.7. Notice that since σi = ci (Q) (as recalled in Section 4.3.7), the Chow ring of the Grassmannian is the generated by the first k Chern classes of the universal quotient bundle Qk . In particular σn−k+1 , . . . , σn vanish if k < n − k, by properties of the Chern Classes.

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Chapter 5

Schubert’s Algebra This chapter is devoted to construct V natural endomorphisms (D1 , D2 , . . .) of the k th exterior power k Mn (k ≥ 1) of a free Zmodule Mn of rank n. The image via Dh (h ≥ 1) of a basis element of Vk Mn can be computed through a Pieri-like formula (Theorem 5.4.1 V and Proposition 5.7.4). This will prove that k Mn is a free module Vk of rank 1 over the subring of EndZ ( Mn ) generated by the Di ’s, isomorphic to the Chow group of the grassmannian variety Gk (V ) (dim(V ) = n), seen as a free module of rank 1 over the Chow ring A∗ (Gk (V )) (Theorem 5.7.6). Since applying Dh to a basis element “simulates” the operation of taking the hth derivative of a (generalized) wronskian (see Chapter 1), the quoted theorem proves indeed that such a differentiation formalism coincides with Schubert calculus. It will be then developed in a purely formal way. The proofs look natural and particularly transparents: for instance, the relations occurring in the presentation of the Chow ring of the Vkgrassmannian, are phrased in terms of “derivatives” vanishing on Mn , while Giambelli’s determinantal formula (necessary to explicitly multiply two arbitrary Schubert cycles in A∗ (Gk (V ))) is seen as a consequence of a more general “integration by parts” formula. 62

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Hasse-Schmidt Derivations on Exterior Algebras

For the time being, let M be any A-module, t an indeterminate over V A and A[[t]] the ring of formal power series in t. Let M [[t]] := V V M ⊗A A[[t]]: an element of M [[t]] is a formal power series with V “coefficients” in M 1 . V 5.1.1 Definition. A Hasse-Schmidt derivation Dt on M is an A-algebra homomorphism: ^ ^ Dt : M −→ M [[t]]. V A Hasse-Schmidt derivation V on M V determines, and is determined by, its coefficients Di : M −→ M , defined via the equality X ^ Dt α = Di (α)ti , ∀α ∈ M. i≥0

5.1.2 Proposition. For each i ≥ 0, Di satisfies (the generalized) Leibniz’s rule: X Dh1 α ∧ Dh2 β, (5.1) Dh (α ∧ β) = h1 +h2 =h h1 ,h2 ≥0

Proof. By explicitly writing that Dt is an A-algebra homomorphism: ^ Dt (α ∧ β) = Dt α ∧ Dt β, ∀α, β ∈ M.

one sees tha Dh (α ∧ β) is the coefficient of th in the expansion of Dt (α ∧ β), which is also the coefficient of th in the expansion of the wedge product (D0 α + D1 α t + D2 α t2 + . . .) ∧ (D0 β + D1 β t + D2 β t2 + . . .), 1 Since the exterior algebra has been defined only for free modules, the reader not familiar with the general definition, may assume that M is free, as will be needed later on. However the construction below holds in general.

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CHAPTER 5. SCHUBERT’S ALGEBRA

i.e. exactly the right hand side of eq. (5.1). In particular, D0 is an algebra homomorphism: D0 (α ∧ β) = D0 (α) ∧ D0 (β), while D1 is a “usual” derivation:

If Dt =

P

D1 (α ∧ β) = D1 α ∧ β + α ∧ D1 β. i≥0

Di ti , the sequence of the “coefficients” D := (D0 , D1 , D2 , . . .)

of Dt will be also said a HS-derivation and the Di ’s will be also said to be the components of D. To denote a HS-derivation the symbols D and Dt shall be used interchangeably. 5.1.3 Proposition. If dt : M −→ M [[t]] is an A-module V homomorphism there is one and only one HS-derivation D : M −→ t V M [[t]] such that Dt|M = dt . V Proof. Since any µ ∈ M is a finite sum of homogeneous elements µk1 + . . . + µkh , we may assume that µ is homogeneous of degree k, Vk i.e. µ ∈ M . Therefore µ is a finite sum products of the form mi1 ∧ . . . ∧ mik . Then one sets: Dt (µ) = Dt (mi1 ∧ . . . ∧ mik ) = dt (mi1 ) ∧ . . . ∧ dt (mik ) getting an extension to the exterior algebra. Suppose now that Ct is another extension such that Ct |M = dt . Then: Ct (mi1 ∧ . . . ∧ mik ) = Ct (mi1 ) ∧ . . . ∧ Ct (mik ) =

= dt (mi1 ) ∧ . . . ∧ dt (mik ) = Dt (mi1 ∧ . . . ∧ mik ), for each k ≥ 0 and each mi1 ∧ . . . ∧ mik ∈ hence unique.

Vk

M . The extension is

V 5.1.4 Definition. A HS-derivation D on M is said to be regular V if D0 ∈ EndA ( M ) is an A-automorphism; a regular HS-derivation is normalized if D0 = idV M .

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The distinction between regular and normalized HS-derivation is immaterial, because to each regular HS-derivation Dt corresponds the normalized one: X V D−1 D0 −1 Di ti . 0 Dt = id M + i≥0

If Dt is a HS-derivation, denote by Dt the map ^ ^ Dt : ( M )[[t]] −→ ( M )[[t]],

defined as: X X X Dt ( αi ti ) = Dt ⊗A 1A[[t]] (αi ti ) = Dt (αi )ti . i≥0

i≥0

(5.2)

i≥0

This is an A[[t]]-algebra endomorphism of

V

M [[t]]. In fact:

  X X X X  Dt ( αi ti ∧ βj tj ) = Dt αi ∧ βj  tk = i

=

X X

k≥0 i+j=k

j

k≥0

k

Dt (αi ∧ βj )t =

X X Dt ( αi ti ) ∧ Dt ( βj tj ). i

i+j=k

X X

k≥0 i+j=k

(Dt αi ∧ Dt βj )tk = (5.3)

j

V V Let HSt ( M ) be V the set of all regular HS- derivations V on M , and let Ct , Dt ∈ HSt ( M ). Define their product in EndA ( M )[[t]] as being: X X Ct ∗ Dt = (Ci ◦ Dj )th (5.4) h≥0 i+j=h

and notice that Ct ∗ Dt (α) = Ct ◦ Dt (α),

∀α ∈

^

M,

by just expanding the two sides of the above equation according to (5.2) and (5.4) V 5.1.5 Proposition. The pair (HSt ( M ), ∗) is a group.

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V Proof. Let Ct , Dt ∈ HSt ( M ). Let us prove that Ct ∗Dt ∈ HSt (M ). In fact Ct ∗ Dt (α ∧ β) =

= = = =

(Ct ◦ Dt )(α ∧ β)) =

Ct (Dt (α) ∧ Ct (β)) = Ct (Dt (α)) ∧ Ct (Ct (β)) =

Ct ◦ Dt (α) ∧ Ct ◦ Dt (β) = Ct ∗ Dt (α) ∧ Ct ∗ Dt (β)

V so that Ct ◦ Dt ∈ HSt ( M ). The composition ◦ is Vobviously associative. The identity 1 := 1M of M V belongs to HSt ( M ). Let D−1 t be the formal inverse of Dt in End( M )[[t]], i.e. −1 V Dt ◦ D−1 t = Dt ◦ Dt = id M ,

existing by the regularity hypothesis on Dt . One has: D−1 t (α ∧ β)

−1 −1 = D−1 t (Dt (Dt )(α) ∧ Dt (Dt )(β)) =

−1 −1 = D−1 t (Dt ((Dt )α ∧ (Dt )β)) = −1 −1 −1 −1 = (D−1 t ◦ Dt )(Dt α ∧ Dt β) = (Dt α ∧ Dt β), V proving that D−1 t ∈ HSt ( M ), which is obviously regular.

5.2

Shift Endomorphisms

5.2.1 From now on M will be a Z-module freely generated by E := (ǫ1 , ǫ2 , . . .): M M := Z · ǫi . i≥0

Any m ∈ M is a unique finite linear combination ments of E. Regard M as a graded Z-module: M M= Mw ,

P

i≥1

mi ǫi of ele-

(5.5)

w≥0

where Mw = Z · ǫ1+w . If m ∈ Mj , the integer wt(m) = j − 1 is said to be the weight of m. For instance, ǫ1 has weight 0, ǫ2 has weight 1 and so on.

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5.2.2 Remarks and Examples. a) The module M will be interpreted, later on, as being the dual of a Zmodule N , freely generated by a basis E := (e1 , e2 , . . .). In this case E will be taken to be the basis of N ∨ dual of E, i.e. such that ǫi (ej ) = δji . This partly motivates the use of superscripts to index the elements of the basis of M . b) Let M := Z⊕M . It is the Z-module freely generated by E = (ǫ0 , ǫ1 , . . . , ), where ǫ0 := 1⊕0M . A model for M is Z∞ , the Z-module of all the functions P : N −→ Z such that {n | P (n) 6= 0} is a finite set. The canonical basis of Z∞ is {δ i |i ≥ 0}, where δ i : N −→P Z is defined by δ i (j) = δji . Any P ∈ Z∞ i can be uniquely written as P = i≥0 P (i)δ . All the free Z-modules of ∞ infinite countable rank are isomorphic to Z , via a unique isomorphism E : Z∞ −→ M , defined by E (δ i ) = ǫi (i ≥ 0). The isomorphism Z∞ −→ M and the basis (ǫi )i≥0 have been indicated with the same letter E, by abuse of notation. c) Take M = Z[X], the Z-module of polynomials with Z-coefficients in one indeterminate. A basis is given by the monic monomials of degree i ≥ 0 : (X n )n≥0 := {1, X, X 2 , X 3 , . . .}. This is another piece of motivation for using superscripts to index the elements of the basis E of M . In this case the module M ⊆ M can be interpreted as the prime ideal (X).

5.2.3 Definition. Let i ≥ 0 be an integer. The ith E-shift endomorphism Di : M −→ M is the unique Z-endomorphism such that Di (ǫj ) = ǫi+j . The shift endomorphism sequence is: D := (D0 , D1 , . . .). Clearly D0 = idM and, for each i > 0, Di is a monomorphism: moreover Di = D1i (agreeing that D10 = D0 ). With respect to the grading (5.5) of M , Di is a homogeneous endomorphism of M of weight (=degree) i, because Di Mj = Mi+j . Let Di M denote the free submodule of M generated by E i := Di E = (ǫi+1 , ǫi+2 , . . .). One has a filtration of M by means of free submodules: DM :

M = D0 M ⊃ D1 M ⊃ D2 M ⊃ . . . .

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Let M ∨ := HomZ (M, Z) be the dual module of M and denote by E := (e1 , e2 , . . .) the basis of M ∨ dual of E, i.e. ei (ǫj ) = δij . 5.2.4 Examples.

a) If M = Z[X], the ith -shift endomorphism of Z[X] (with respect to the basis (1, X, . . .)) is the multiplication by X i . In fact X i · X j = X i+j . Also in this case one has X i = X1i . b) Let Iδ = (−δ, δ) ⊂ R, with 0 < δ < 1 and f : Iδ −→ R be the function f (x) =

1 . 1−x

Then f is a function of class C ∞ on Iδ . Let f (i) (x) be the i-th derivative of f . Since Ci , Ci ∈ R f (i) (x) = (1 − x)i+1 then (f, f ′ , f ′′ , . . .) are linearly independent in the R-vector space C ∞ (Iδ ), L (i) and hence also linearly independent over Z. Define M as . i≥0 Z · f i i (j) (i+j) Then, the shift operator is precisely d /dx sending f (x) onto f (x). This kind of examples inspired the choice of the symbol Di (as for derivative) for the ith -shift operator and again the superscripts to denote the elements of the basis E of M .

5.2.5 The ring A∗ (M, D). If λ = (1m1 2m2 . . .) is a partition, define: Dλ = D1m1 ◦ . . . ◦ Dkmk ∈ EndZ (M ), where ◦ is the composition in EndZ (M ) and Dij = Di ◦ . . . ◦ Di , {z } | j times

is the j th iterated of Di . Let evD : Z[T] −→ EndZ (M ) be the module homomorphism defined by: Tλ 7−→ evD (Tλ ) := Dλ . The image is the Z-submodule of EndZ (M ):   X  A∗ (M, D) := aλ Dλ | L is a finite subset of P .   λ∈L

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69

If P (T) ∈ Z[T], we shall simply write P (D) instead of evD (P (T)).

5.2.6 Proposition. The set A∗ (M, D) is the minimum Z-subalgebra of EndZ (M ) containing all the Di ’s. Moreover it is commutative. Proof. The fact that A∗ (M, D), with respect to the composition of endomorphisms, is a Z-subalgebra of EndZ (M ) is obvious; it is commutative because Di Dj = Dj Di , for all i, j ≥ 0. Suppose now that A is any Z-subalgebra containing all the Di ’s (i ≥ 0). Then it contains Dλ , too, for each λ ∈ P, and therefore any finite Z-linear combination of them. Then A ⊃ A∗ (M, D).

Because of the commutativity of A∗ (M, D), the natural evaluation epimorphism: evD : Z[T] −→ A∗ (M, D),

defined by evD (Tλ ) = Dλ , extends to a ring homomorphism, again denoted by the same symbol evD . It can be defined on the algebra generators by evD (Ti ) = Di . Then A∗ (M, D) is itself a graded ring: M A∗ (M, D) = Ah (M, D), h≥0

where Ah (M, D) = evD (Z[T]h ). Let evǫ1 : A∗ (M, D) −→ M . P (D) 7−→ P (D) · ǫ1 Clearly evǫ1 is surjective. In fact for any m = a1 ǫ1 + a2 ǫ2 + . . .+ an ǫn , there is Gm (D) ∈ A∗ (M, D) such that Gm (D)ǫ1 = m: it suffices to choose Gm (D) = a1 idM + a2 D1 + . . . + an−1 Dn . Furthermore evD,ǫ1 := evǫ1 ◦ evD : Z[T] −→ M, is an epimorphism and the equality ker(evD,ǫ1 ) = ker(evD ) holds. In fact ker(evD ) ⊆ ker(evD,ǫ1 ), by the very definition of evD,ǫ1 . Conversely, if P (T) ∈ ker(evD,ǫ1 ), then evD (P (T))ǫ1 = 0M , i.e. for each m ∈ M : evD (P (T))m = evD (P (T)) · Gm (D)ǫ1 = Gm (D)evD (P (T))ǫ1 = 0,

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meaning that evD (P (T)) = 0A∗ (M,D) , i.e. P (T) ∈ ker(evD ). This proves that: 5.2.7 Proposition. There are Z-module isomorphisms: evD

Z[T] ker(evD )

−→ A∗ (M, D)

evǫ1

−→ M.

Next task consists in determining explicitly ker(evD ), but this is easy. Since Di = D1i , the ideal (Ti − T1i )i≥1 of Z[T] is contained in the kernel, whence the surjection: Z[T1 ] ∼ =

(T2 −

Z[T] Z[T] ∼ −→ = M, 3 − T1 , . . .) ker(evD )

T12 , T3

T1i−1

given by 7→ ǫi . This latter map Pn is clearly injective (if evD (a0 + i1 a1 T + . . . + an T in ) = 0, then j=1 aij ǫij +1 = 0, i.e. all aij = 0) and then ker(evD ) = (Ti − T1i )i≥0 and Z[T1 ] ∼ = A∗ (M, D) ∼ = M.

5.2.8 A few words about the “extended” module M = Z ⊕ M . First of all the shift endomorphisms Di : M −→ M clearly extend to shift endomorphisms Di : M −→ M (by abuse denoted by the same letter), by defining Di ǫ0 = ǫi . The corresponding filtration is then: DM :

M = D0 M ⊃ D1 M ⊃ D2 M ⊃ . . .

extending (5.2.1). In particular one has D1 M = M . In this case one has again A∗ (M , D) ∼ = Z[T1 ] and the isomorphism A∗ (M , D) −→ M

is given by sending D1n 7→ ǫn = evD,ǫ0 (T1n ).

5.3

Schubert Derivations

5.3.1 If D := (D0 , D1 , . . .) is the E-shift sequence of M , denote by: X Dt := Di ti = idM + D1 t + D2 t2 + . . . ∈ EndZ (M )[[t]], i≥0

its corresponding shift formal power-series. Let algebra of M .

V

M be the exterior

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71

5.3.2 Definition. The Hasse-Schmidt extension of Dt to said to be Schubert derivation (S-derivation, for short).

V

M is

The S-derivation is then the unique Z-algebra homomorphism ^ ^ Dt : M −→ M [[t]],

P such that Dt (ǫi ) = j≥0 ǫi+j tj . The algebra homomorphism V condition V means that Dt (α ∧ β) = Dt α ∧ Dt β, for each α, β ∈ M . If α ∈ M and h ≥ 0, Dh α will be said to be the hth S-derivative of α. In particular V one has that D0 is the identity and that, see 5.1.2, Dh ∈ EndZ ( M ) satisfies Leibniz’s rule (5.1). V For each h ≥ 0, Dh is a homogeneous endomorphism of M of degree 0 with respect to the grading ^

M :=

k M^

M,

k≥0

Vk Vk i.e. Dh ( M ) ⊆ M . By abuse, the induced endomorphism Di |V k M : will be still denoted by Di .

k ^

M −→

k ^

M

5.3.3 Proposition. The endomorphisms Di : (i ≥ 0), are pairwise commuting.

Vk

M −→

Vk

M

Proof. By induction on k. For k = 1 the claim is true by construcVk tion. Assume that the property holds for k − 1. Since any m ∈ M is a V finite sum of k-vectors of the form α ∧ β, with α ∈ M and k−1 β∈ M , without loss of generality one may check the property for any m of this form. Then: X Dj1 α ∧ Dj2 β) = Di Dj (α ∧ β) = Di ( j1 +j2 =j

=

X

X

i1 ,i2 ≥0

j1 ,j2 ≥0

j1 ,j2 ≥0

Di1 Dj1 α ∧ Di2 Dj2 β.

(5.6)

i1 +i2 =i j1 +j2 =j

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By the inductive hypothesis, last member of equality (5.6) is equal to: X X X Dj1 Di1 α ∧ Dj2 Di2 β = Dj ( Di1 α ∧ Di2 β) = j1 +j2 =j i1 +i2 =i

i1 +i2 =i

j1 ,j2 ≥0

i1 ,i2 ≥0

i1 ,i2 ≥0

=

Dj Di (α ∧ β).

As a consequence, the natural evaluation Z-module homomorphism  V  evD : Z[T] −→ EndZ ( k M ) 

Tλ

Dλ

7−→

V V maps Z[T] onto a commutative subring A∗ ( k M , D) of EndZ ( k M ). Vk It turns out that for k > 1, as already for k = 1, M is itself a graded Z-module with respect to the weight grading defined below.

5.3.4 Definition. The weight of a non zero element ǫi1 ∧ . . . ∧ ǫik ∈ Vk M is the weight of the k-schindex I: wt(ǫi1 ∧ . . . ∧ ǫik ) = (i1 − 1) + . . . + (ik − k) =

k X j=1

ij −

k(k + 1) . 2

If I = (1 ≤ i1 < . . . < ik ), the weight of ǫi1 ∧ . . . ∧ ǫik coincides with the weight of the associated partition λ(I) = (ik − k, . . . , i1 − 1). Notice also that wt(ǫi1 ∧ . . . ∧ ǫik ) = wt(ǫiτ (1) ∧ . . . ∧ ǫiτ (k) ), Vk Vk for all τ ∈ Sk . Let ( M )w be the submodule of M spanned by all ǫi1 ∧ . . . ∧ ǫik of weight w. Then, clearly: ! ! ! ! k k k k k ^ ^ ^ ^ M ^ M= M ⊕ M ⊕ M ⊕ ... = M , 0

and this makes

Vk

1

2

w≥0

w

M into a graded Z-module.

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5.3.5 Remark on the terminology. The terminology weight has been suggested by the theory of Weierstrass points. In fact the Weierstrass gap sequence of a point P of a smooth curve C is a g-schindex 1 = n1 < . . . < ng , and its weight is precisely (n1 − 1) + . . . + (ng − g). In his thesis [59], Ponza shows that if 1 < n2 < . . . < ng is the Weierstrass gap sequence at P , then the generalized wronskian section ω ∧ Dn1 −1 ω ∧ . . . ∧ Dng −1 ω. does not vanishes at P .

V V Proposition 5.3.6 below shows that Dh : k M −→ k M is inVk deed a homogeneous endomorphism of degree (=weight) h of M. 5.3.6 Proposition. The “generalized” Leibniz’s rule holds: X Dh (ǫi1 ∧ ǫi2 ∧ . . . ∧ ǫik ) = ǫi1 +h1 ∧ ǫi2 +h2 ∧ . . . ∧ ǫik +hk , h1 +...+hk =h hi ≥0

for each h ≥ 0.

Proof. By induction on k. If k = 1 the property is obviously true. Assume it holds for k − 1. Then i1

i2

ik

Dh (ǫ ∧ ǫ ∧ . . . ∧ ǫ ) =

h X

h1 =0

ǫi1 +h1 ∧ Dh−h1 (ǫi2 ∧ . . . ∧ ǫik ). (5.7)

By the inductive hypothesis: Dh−h1 (ǫi2 ∧ . . . ∧ ǫik ) =

X

h2 +...+hk =h−h1

ǫi2 +h2 ∧ . . . ∧ ǫik +hk ,

and the r.h.s. of formula (5.7) turns into: P Dh (ǫi1 ∧ ǫi2 ∧ . . . ∧ ǫik ) = h1 +...+hk =h ǫi1 +h1 ∧ . . . ∧ ǫik +hk .

Vk

In particular evD (Tλ ) := Dλ is a homogeneous endomorphism of M of weight |λ|, i.e.: k k ^ ^ Dλ ( M )w ⊆ ( M )w+|λ| .

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5.3.7 Example. It is of course not necessary to remember formula (5.7) by head, since it is just the formalization of a V mechanical practical rule. For instance, let us compute D2 (ǫ2 ∧ ǫ3 ∧ ǫ5 ) ∈ 3 M . One has:

D2 (ǫ2 ∧ ǫ3 ∧ ǫ5 ) = D2 ǫ2 ∧ ǫ3 ∧ ǫ5 + D1 ǫ2 ∧ D1 (ǫ3 ∧ ǫ5 ) + ǫ2 ∧ D2 (ǫ3 ∧ ǫ5 ) = = ǫ4 ∧ ǫ3 ∧ ǫ5 + ǫ3 ∧ (ǫ4 ∧ ǫ5 + ǫ3 ∧ ǫ6 ) + ǫ2 ∧ (ǫ5 ∧ ǫ5 + ǫ4 ∧ ǫ6 + ǫ3 ∧ ǫ7 ) = = ǫ4 ∧ ǫ3 ∧ ǫ5 + ǫ3 ∧ ǫ4 ∧ ǫ5 + ǫ3 ∧ ǫ3 ∧ ǫ6 + ǫ2 ∧ ǫ5 ∧ ǫ5 + ǫ2 ∧ ǫ4 ∧ ǫ6 + +ǫ2 ∧ ǫ3 ∧ ǫ7 = ǫ2 ∧ ǫ4 ∧ ǫ6 + ǫ2 ∧ ǫ3 ∧ ǫ7 , where last equality results from the vanishing of the terms having equal ∧-factors and to the cancellations due to skew-symmetry.

5.4

Pieri’s Formula for S-Derivations

Example 5.3.7 shows that computing S-derivatives of k-vectors is a straightforward matter and that one has to care only about possible vanishing and cancellations. However, the practice of many examples naturally suggests the following theorem which says that our algebraic model is isomorphic to Schubert Calculus (Section 4.3). 5.4.1 Theorem. Let I be a k-schindex. Then Pieri’s formula for S-derivatives holds: X ǫi1 +h1 ∧ . . . ∧ ǫik +hk , (5.8) Dh (ǫi1 ∧ . . . ∧ ǫik ) = (hi )∈H(I,h)

h1 +...+hk =h

where H(I, h) is the set of all k-tuples (hi ) ∈ Nk such that: 1 ≤ i1 ≤ i1 + h1 < i2 ≤ i2 + h2 < . . . ≤ ik−1 + hk−1 < ik

(5.9)

and h1 + h2 + . . . + hk = h. Proof. (It is copied from [23]) By induction on the integer k. For k = 1, formula (5.8) is trivially true. Let us prove it directly for k = 2. For each h ≥ 0, let us split sum (5.7) as: X ǫi1 +h1 ∧ ǫi2 +h2 = P + P, Dh (ǫi1 ∧ ǫi2 ) = (5.10) h1 +h2 =h h1 ,h2 ≥0

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75

where: X

P=

ǫi1 +h1 ∧ ǫi2 +h2

and P =

X

ǫi1 +h1 ∧ ǫi2 +h2 .

i1 +h1 1. Then one proves the following: 5.5.7 Vk Lemma. For all k > 0 and all h > k, ∆h α = 0 for all α∈ M , while: ∆k (ǫi1 ∧ . . . ∧ ǫik ) = ǫi1 +1 ∧ . . . ∧ ǫik +1 .

(5.21)

Proof. If h > k, each summand of the r.h.s. of (5.20) possesses at least an index 1 ≤ i ≤ k, such that hi > 1. On the other hand, if in the sum (5.20) h = k, only the term for which h1 = . . . = hk = 1 survives, i.e. precisely the r.h.s. of (5.21) 5.5.8 Corollary. Giambelli’s problem has a solution for ǫn−k+1 ∧ . . . ∧ ǫn . Proof. Let ∆k ∈ Z[T] be the polynomial such that evD (∆h (T)) = ∆k (D). It is sufficient to take Gn−k+1,...,n = (∆k )k(n−k) . In fact: evD (∆k (T)k(n−k) )ǫ1 ∧ . . . ∧ ǫk = evD (∆k (T))k(n−k) · ǫ1 ∧ . . . ∧ ǫk =

= ∆k (D) ◦ . . . ◦ ∆k (D) ǫ1 ∧ . . . ∧ ǫk = ǫn−k+1 ∧ . . . ∧ ǫn . | {z } k(n−k) times

We can hence prove the most important result of this section: 5.5.9 Theorem. The map evD,ǫ1 ∧...∧ǫk : Z[T] −→ is surjective for all k.

k ^

M

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81

Proof. It is sufficient to prove that Giambelli’s problem has solution for each k-schindex I = (i1 , . . . , ik ). By Corollary 5.5.5, there is a polynomial P ∈ Z[T] such that: ǫi1 ∧ . . . ∧ ǫik = P (D)ǫi1 ∧ ǫi1 +1 ∧ . . . ∧ ǫi1 +k−1 .

Then, applying 5.5.8: ǫi1 ∧ . . . ∧ ǫik = P (D) · ∆k (D)k(i1 −1) · ǫ1 ∧ . . . ∧ ǫk , proving our claim. 5.5.10 Corollary. The following maps are isomorphisms: k evD evǫ1 ∧...∧ǫk ^ Z[T] −→A∗ (∧k M, D) −→ M, (ker(evD,ǫ1 ∧...∧ǫk ))

where by abuse of notation one denotes by evD the induced map on the quotient. Proof. The surjectivity of evD,ǫ1 ∧...∧ǫk and of evD implies the surjectivity of evǫ1 ∧...∧ǫk . One must only show that ker(evD ) = ker(evD,ǫ1 ∧...∧ǫk ), and for this one argues exactly as in the proof of Proposition 5.2.6. Next section shall be devoted to find Vkan explicit presentation in terms of generators and relations of A∗ ( M , D).

5.6

A Presentation for A∗ (

Vk

M , D)

Up to translations, most of the preliminary propositions of this Section can be found in [52], there proven within the language of symmetric functions. 5.6.1 Definition. A Giambelli function G : I k −→ Z[T] is a function which associates to each k-schindex I = (i1 , . . . , ik ) a Giambelli polynomial for ǫi1 ∧ . . . ∧ ǫik . Its image G will be said a Giambelli’s set.

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5.6.2 Corollary. Any Giambelli’s set is a Z-basis for Z[T]. Proof. First of all we prove P that G is a linearly independent subset of elements of Z[T]. Let λ∈I ′ aI [GI (T)] = 0 be any non trivial linear dependence relation, where I ′ is a finite subset of I. There is then k ≥ 1, such that I ′ ⊆ I k and X aI GI (D)ǫ1 ∧ . . . ∧ ǫk = 0, I∈I ′

which is impossible, because Gi1 ...ik (D)ǫ1 ∧ . . . ∧ ǫk = ǫi1 ∧ . . . ∧ ǫik Vk are all linearly independents, being part of a basis of M. Let us now prove that they are Z-generators of Z[T]. Let G h = {GI ∈ G | wt(I) = h}. It is sufficient to show that G h is in fact a Zbasis for Z[T]h . To this purpose, notice that GI (T) ∈ Gh is a unique Z-linear combination of T λ ’s, (λ ∈ Ph ): X

GI (T) =

aλI DλI .

λI ∈Ph

Then, since ♯{G h (T)} = ♯{T µ : µ ∈ P h } (because of the bijection between k-schindices and partitions of length at most k, see Sect. 2.1.6), and all the GI ∈ G h are linearly independent, they freely span Q[T]h = Z[T]h ⊗Z Q. In particular, for each λ ∈ Pk , T λ is a unique Q-linear combination of GI (T) ∈ G h (T): Tλ =

X

bIλ GIλ (T).

(5.22)

Iλ

We only need to show that the coefficients bIλ ∈ Z. But equality (5.22) implies: evD (T λ )ǫ1 ∧ . . . ∧ ǫk =

X

Iλ ∈Ih

bIλ evD (GIλ (T))ǫ1 ∧ . . . ∧ ǫk

(5.23)

The l.h.s. is a Z-linear combination of ǫi1 ∧ . . . ∧ ǫik : X aIλ ǫi1 ∧ . . . ∧ ǫik . Iλ ∈Ph

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By unicity, the coefficients bIλ ’s must be the same as those occurring in the expansion of the l.h.s. of (5.23) as linear combination of of the ǫi1 ∧ . . . ∧ ǫik , i.e. they must be integers. 5.6.3 Proposition. Let φ ∈ Z[T1 , . . . , Tk ] ⊆ Z[T] such that φ(D1 , . . . , Dk ) · ǫ1 ∧ . . . ∧ ǫk = 0, then φ = 0. Proof. Let φ ∈ Z[T1 , . . . , Tk ] of degree h. Writing φ as the sum φ0 + φ1 + . . . + φh , where φi is homogeneous of degree i, one has k ^ φi (D)ǫ1 ∧ . . . ∧ ǫk ∈ ( M )i .

Therefore φ(D)ǫ1 ∧ . . . ∧ ǫk = 0 if and only if φi (D)ǫ1 ∧ . . . ∧ ǫk = 0, for each homogeneous summand of φ. One may then assume that φ ∈ Z[T]h (i.e. it is homogeneous of weight h). Since Gh is a Z-basis for Z[T]h , one may write: X φ(T1 , . . . , Tk ) = aI GI (T). I∈I k

Hence: 0 = φ(D1 , . . . , Dk )ǫ1 ∧ . . . ∧ ǫk =

X

I∈I k

aI GI (D)ǫ1 ∧ . . . ∧ ǫk ,

implying that all aI = 0, because {GI (D) · ǫ1 ∧ . . . ∧ ǫk }I∈I k are linearly independent. We may now prove the following result computing explicitly the presentation of A∗ (M, D).

5.6.4 Theorem. The ring A∗ (M, D) is the polynomial ring Z[D1 , . . . , Dk ]. (Cf. [7]) Proof. By Proposition 5.6.3, D1 , . . . , Dk are algebraically independents. Moreover Dk+i (i ≥ 1) is a Z-polynomial in D1 , . . . , Dk . In fact, for each j ≥ 0, the relation (1/Dt )·Dt = Dt ·(1/Dt ) = 1 implies: Dk+i − ∆1 Dk+i−1 + . . . + (−1)k+i−1 ∆k+i−1 D1 + (−1)k+i ∆k+i = 0. (5.24)

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Since ∆p is a polynomial expression in D1 , . . . , Dp only, it follows that (for i = 1) Dk+1 is a polynomial in D1 , . . . , Dk , because ∆k+1 = 0 by Lemma 5.5.7. Suppose now that the property holds for all 1 ≤ j ≤ i − 1. Then formula (5.24) reads as: Dk+i = −∆1 Dk+i−1 + . . . + (−1)k+1 ∆k Di ,

(5.25)

because ∆k+i = 0 for all i ≥ 1; it follows that Dk+i is a polynomial in D1 , . . . , Dk , by induction. As a conclusion: k ^ A∗ ( M , D) =

5.7

Z[D1 , D2 , . . .] ∼ = Z[D1 , . . . , Dk ]. (Dk+i − Dk+i (D1 , . . . , Dk ))i≥1

The Finite Case

Let Dn M be the submodule of M spanned by E n = (ǫn+1 , ǫn+2 , . . .) and denote by Mn the free submodule of M generated by En = (ǫ1 , . . . , ǫn ). Let pn be the projection map:  Mn pn : M −→ 

m

7−→

Pn

i=1 ei (m)ǫ

i

and ιn : Mn −→ M be the section associated to pn , defined by ιn (m) = m, for each m ∈ Mn . One has the sequence: 0 −→ Dn M −→ M −→ Mn −→ 0, which can easily be checked to be exact. 5.7.1 Proposition. The natural epimorphism: pn ◦ evǫ1 : Z[T1 ] −→ Mn induces the isomorphism: Z[T1 ] ∼ = Mn (T1n )

(5.26)

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Proof. From the epimorphism: pn ◦ evD,ǫ1 : Z[T1 ] −→ Mn one gets Z[T1 ]/ ker(pn ◦ evD,ǫ1 ) ∼ = Mn . Now, pn (D1i ǫ1 ) = 0 if and only if i ≥ n, proving that the kernel of pn ◦ evD,ǫ1 is precisely (T1n ). This proves the claim. 5.7.2 Remark. Notice that the l.h.s. of (5.26) is isomorphic to the Chow ring of Pn−1 . Denote by A∗ (Mn , D) the image of Z[T1 ]/(T1n ) in EndZ (Mn ). Then A∗ (Mn , D) is isomorphic to Z[D1 ] where D1 may be identified with the class of T1 mod (T1n ). Let now

^

M ∧ Dn M V be the ideal of the exterior algebra M generated byVDn M : a typical element of it is of the form α ∧ ǫj , with j > n, α ∈ M . Let k−1 ^

M ∧ Dn M :=

k ^

^ M ∩ ( M ∧ Dn M ).

Vk It is the submodule of M generated by all ǫi1 ∧ . . . ∧ ǫik such that ik > n. We claim that: k ^

Vk M Mn := Vk−1 M ∧ Dn M

In fact, one has a surjective module homomorphism: ∧k pn : defined by ∧k pn (

X

1≤i1 0): D15 (ǫ1 ∧ ǫ3 ∧ ǫ6 ) = D15 (ǫ1 ∧ ǫ3 ) ∧ ǫ6 = D14 (ǫ2 ∧ ǫ3 + ǫ1 ∧ ǫ4 ) ∧ ǫ6 = = D13 (2ǫ2 ∧ ǫ4 + ǫ1 ∧ ǫ5 ) ∧ ǫ6 = D12 (ǫ3 ∧ ǫ4 + 3ǫ2 ∧ ǫ5 ) ∧ ǫ6 = = D1 (5ǫ3 ∧ ǫ5 ) ∧ ǫ6 = 5 · ǫ4 ∧ ǫ5 ∧ ǫ6 . Since [ǫ4 ∧ ǫ5 ∧ ǫ6 ] is the class of the point in A∗ (G(3, 6)), the solution of our enumerative exercise is 5 as claimed in Example 4.3.9. 6.2.4 Example. Here we concludes the computation of the product (σ12 − σ2 ) · σ21 proposed in Example (4.3.11). It is the same as computing: (D12 − D2 )(ǫ1 ∧ ǫ3 ∧ ǫ5 ). We left to the Reader the task of computing directly the above expression. Instead, we use the fact that, indeed, D12 − D2 = ∆2 (see Lemma 5.5.7), and that ∆2 enjoy Leibniz’s rule for second derivatives. One has: ∆2 (ǫ1 ∧ ǫ3 ∧ ǫ5 ) = ∆2 (ǫ1 ∧ ǫ3 ) ∧ ǫ5 + ∆1 (ǫ1 ∧ ǫ3 ) ∧ ∆1 ǫ5 = where we used again Lemma 5.5.7 for concluding that ∆2 (ǫ5 ) = 0. Thus, continuing computations and remembering that ∆1 = D1 : = ǫ2 ∧ ǫ4 ∧ ǫ5 + ǫ2 ∧ ǫ3 ∧ ǫ6 + ǫ1 ∧ ǫ4 ∧ ǫ6 = (σ221 + σ311 + σ32 ) ∩ [G(3, 6)], having used, in the last equality, the dictionary translating the expression ǫi1 ∧ . . . ∧ ǫik into the corresponding σλ(I) .

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99

Degree of Grassmannians

6.3.1 Proposition. Newton’s binomial formula holds: D1n (α

n X n ∧ β) = D1k α ∧ D1n−k β, k

(6.6)

k=0

V for each α, β ∈ M . Proof. For n = 1 the formula is obvious. Suppose it holds for n − 1. Then D1n (α

∧ β) =

D1 (D1n−1 (α

∧ β)) = D1

n−1 X k=0

=

n−1 X k=0

=

n−1 X k=0

=

n X

k=1

=

n−1 X k=1

=

n X

k=0

!

! n−1 D1k α ∧ D1n−1−k β = k

n−1 D1 D1k α ∧ D1n−1−k β = k ! ! n−1 X n−1 n−1 k+1 n−1−k D1 α ∧ D1 β+ D1k α ∧ D1n−k β = k k k=0 ! ! n−1 X n−1 n−1 D1k α ∧ D1n−k β + D1k α ∧ D1n−k β = k−1 k k=0 ! !! n−1 n−1 + D1k α ∧ D1n−k β + D1n α ∧ β + α ∧ D1n β = k−1 k ! n D1k α ∧ D1n−k β. k “

”

where in the first row we used induction in the last equality. The rest are straightforward manipulations.

6.3.2 The degree of the grassmannian G(k, n) in the Pl¨ ucker embedding is, by definition: Z k(n−k) dk,n = σ1 ∩ [G(k, n)], i.e. it is the coefficient of the generator of ∧k Mn of maximum weight: k(n−k) 1

D1

ǫ ∧ . . . ∧ ǫk = dk,n · ǫn−k+1 ∧ . . . ∧ ǫn

(6.7)

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CHAPTER 6. MISCELLANEA

The reason is that σ1 ∩ [G(k, n)] is a very ample divisor class emn bedding the Grassmannian in P(k )−1 à la Pl¨ ucker. The degree of the Pl¨ ucker image of G(k, n) is therefore the degree of its intersection with as many hyperplanes as the dimension of the grassmannian. The pullback of the hyperplane class of P l(G(k, n)) is precisely σ1 and this explains formula (6.7). 6.3.3 Example (Degree of G(2, n)). To compute the degree d2,n of G(2, n) is then sufficient to expand expression: 2(n−2)

D1

(ǫ1 ∧ ǫ2 )

which is certainly an integral multiple of ǫn−1 ∧ ǫn . First we apply the binomial formula: 2(n−2) 1 D1 (ǫ

2

∧ǫ )=

2(n−2)

X

k=0

2(n − 2) 2(n−2)−k 2 ǫ . D1k ǫ1 ∧ D1 k

Vk 2(n−1) 1 Now D1 ǫ ∧ ǫ2 ∈ ( Mn )2(n−2) . The latter is a free Z-module spanned by ǫn−1 ∧ ǫn only. Therefore one knows a priori that all the summands cancel and that only a Z-linear combination of ǫn−1 ∧ ǫn and ǫn ∧ ǫn−1 will survive. More precisely one is interested in the sum: 2(n − 2) 2(n−2) 1 2(n−2)−n+2 2 D1 (ǫ ∧ ǫ2 ) = D1n−2 ǫ1 ∧ D1 ǫ + n−2 2(n − 2) 2(n−2)−n+1 2 + D1n−1 ǫ1 ∧ D1 ǫ = n−1 2(n − 2) 2(n − 2) = − ǫn−1 ∧ ǫn . n−2 n−1 Hence: d2,n =

2(n − 2) 2(n − 2) (2n − 4)! − = . n−2 n−1 (n − 1)!(n − 2)!

6.3.4 Example. Working on the full exterior algebra, instead of on a single exterior power, suggests that Schubert calculus for grassmannians G(k, n) must be linked recursively with Schubert calculus

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of G(k ′ , n′ ) with k ′ < k and n′ ≤ n. This is also geometrically observed in [27]. Some nice exercises can be done to this purpose and, as a matter of example, we look for a formula expressing d3,n in functions of degrees of G(2, n − 1) and G(2, n). As before, one aims to determine the integer d3,n defined by the equality below: 3(n−3) 1

ǫ ∧ ǫ2 ∧ ǫ3 = d3,n ǫn−2 ∧ ǫn−1 ∧ ǫn

D1

To this purpose one first uses Newton bynomial formula (6.6): 3(n−3) 1 D1 ǫ

2

3

∧ǫ ∧ǫ =

3(n−3)

X

k=0

3(n − 3) 3(n−3)−k 1 D1 (ǫ ∧ ǫ2 ) ∧ D1k ǫ3 k

In the above sum will only survive multiples of the three terms below: ǫn−2 ∧ ǫn−1 ∧ ǫn ,

ǫn−1 ∧ ǫn ∧ ǫn−2 ,

ǫn−2 ∧ ǫn ∧ ǫn−1 .

One then selects in the above sum those summands for which the values of k are n − 3, n − 2 and n − 1. In other words: 3(n − 3) 3(n−3) 1 2(n−3) 1 D1 ǫ ∧ ǫ2 ∧ ǫ3 = D1 (ǫ ∧ ǫ2 ) ∧ ǫn + n−3 3(n − 3) + D12n−5 (ǫ1 ∧ ǫ2 ) ∧ ǫn−1 + n−4 3(n − 3) 2(n−2) 1 + D1 (ǫ ∧ ǫ2 ) ∧ ǫn−2 . n−5 To end computations we only need to know the coefficients of 2(n−3)

(ǫ1 ∧ ǫ2 )

ǫn−2 ∧ ǫn−1

in the expansion of

D1

ǫn ∧ ǫn−2

in the expansion of

D12n−5 (ǫ1 ∧ ǫ2 )

ǫn−1 ∧ ǫn

in the expansion of

D1

2(n−2)

(ǫ1 ∧ ǫ2 )

By the previous discussion, the first and the third coefficient are nothing else than d2,n−1 and −d2,n respectively. As for the second coefficient, remark that the coefficient of ǫn ∧ ǫn−2 in the expansion

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of D12n−5 (ǫ1 ∧ ǫ2 ) is the same as the coefficient of ǫn ∧ ǫn−1 in the expansion of D12n−4 (ǫ1 ∧ǫ2 ) and this latter is precisely the degree d2,n of the Grassmannian G(2, n). Therefore we have proven the following formula: 3(n − 3) 3(n − 3) 3(n − 3) d3,n = d2,n−1 + − d2,n . n−3 n−5 n−4 In an attempt to recognize a general pattern, Taise Santiago computed a similar recursive formula for d4,n : 4(n − 4) 4(n − 4) 4(n − 4) d4,n = d3,n−1 + − d3,n + n−4 n−6 n−7 4(n − 4) 3n − 11 3n − 11 + − d2,n + n−5 n−5 n−6 3n − 11 3n − 11 − d2,n−1 − d2,n−2 . n−5 n−3

6.4

Wronskians Correspondences

A complete flag of V is a filtration: E • : E 0 ⊃ E 1 ⊃ E 2 ⊃ . . . ⊃ E n = (0)

(6.8)

into subspaces E i of V of codimension i (hence E 0 = V ). Let F l(V ) be the set of all the complete flags of the complex vector space V . It is a honest smooth connected complex projective variety of dimension n 2 , intimately related with any grassmannian Gk (V ), 1 ≤ k ≤ n. It is easy to guess that a k-plane [Λ] is in general position with respect to a flag E • if and only if E k ∩[Λ] = (0). In fact one has the following chain of inequalities: k = dim(E 0 ∩ [Λ]) ≥ dim(E 1 ∩ [Λ]) ≥ . . . ≥ dim(E k ∩ [Λ]) = 0, implying that dim(E i ∩ [Λ]) = k − i for i ≤ k and E i ∩ [Λ] = 0 for all i > k. That is, [Λ] is in general position with respect to the flag (6.8) if it is in general position with respect to all subspaces defining it. Such a regular behaviour gets suddenly lost for all k-planes such that: dim([Λ] ∩ E k ) > 0,

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i.e. those we call E • -special k-planes – k-planes in special position with respect to the flag E • . Similarly, pick a k-plane [Λ] in V . Then the general complete flag of V F • : F 0 ⊃ F 1 ⊃ F 2 ⊃ . . . ⊃ F n = (0) has the property that [Λ] is not F • -special: it suffices to pick a general (n − k)-plane, intersecting [Λ] only at the zero vector, and then completing it in a flag around it. Hence the geometry of the flag varieties and that of grassmannians someway interacts as suggested by the following diagram

Fl(V) ✕ Gk(V) pr 1

π1

Fl(V)

∪ Wk

π2

pr 2

G k(V)

where Wk := {([Λ], F • ) ∈ F l(V ) × Gk (V ) | [Λ] is F• − special}. Indeed, Schubert calculus for Grassmannian studies the cohomology (or the intersection theory) of fibers of the map π1 , while the so called Schubert Calculus for complete Flag Varieties (see e.g. [22], [75]) studies the cohomology of the fibers of π2 . 6.4.1 The smooth irreducible algebraic variety Wk is what it would be reasonable to call a Wronskian Correspondence. In fact the fiber of Wk over E • is the zero locus of a section WE of the line bundle ∧k Tk∨ . Moreover this section looks like a true wronskian (as shown in Example 4.2.7) and one has that [Λ] ∈ π1−1 (E • ) if and only if ǫ1 ∧ . . . ∧ ǫk (Λ) = 0, where (ǫ1 , ǫ2 , . . . , ǫn ) is a basis of V adapted to the flag E • . 6.4.2 Example (Weierstrass Points on Curves). This example has been the main inspiration of these notes. The reader not familiar with elementary theory of algebraic curves may skip it with no essential loss

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or consult [24] for a friendly introduction. Let C be a smooth projective complex curve and let K be its canonical bundle. The space H 0 (C, K) of holomorphic differentials has dimension g over the complex field. Denote by J i K the ith jet extension of K: it is a rank i + 1 bundle whose sections behave as do i-th derivatives of local sections of K. If ω ∈ H 0 (K), then Di ω is a section of J i K which locally is the local representation of ω together with its first i-derivatives. A point P is said to be a Weierstrass point if there exists a non zero ω ∈ H 0 (K) such that Dg−1 (ω)(P ) = 0, i.e. if there exists a non zero holomorphic differential vanishing at P with multiplicity at least g. Hence a Weierstrass point is in the locus where the map Dg−1 : H 0 (K) ⊗ OC −→ J g−1 K defined by Dg−1 (P, ω) = (Dg−1 ω)(P ) drops rank. This is equivalent to look at the zero locus of the determinant map: k ^

Dg−1 : OC −→

g ^

⊗ J g−1 K ∼ =K

g(g+1) 2

,

said to be wronskian. If ω = (ω1 , . . . , ωg ) is a basis of H 0 (K) and U is an open set of C (in the Zariski or usual topology) trivializing K , where ωi|U = ui (z) · dz (z being a coordinate on U ), then (∧k Dg−1 )|U

=

=

g(g+1)

(u ∧ u′ ∧ . . . ∧ u(g−1) ) · dz ⊗ 2 = ˛ ˛ ˛ u1 u2 ... ug ˛ ˛ ′ ˛ ˛ u1 u′2 ... u′g ˛ g(g+1) ˛ ˛ ⊗ 2 . ˛ .. .. .. ˛ · dz . . ˛ . . . . ˛˛ ˛ ˛u(g−1) u(g−1) . . . u(g−1) ˛ g 1 2

Abusing notation the wronskian section is also written as ω ∧ Dω ∧ . . . ∧ Dg−1 ω.

Let (U, z) be a trivialization around a point P . Then the points of the fiber JPi K of J i K at P are (i + 1)-tuples of complex numbers (u0 , u1 , . . . , ui ). Consider the map JP2g−2 K −→ JPi K defined by (u0 , u1 , . . . , ui , ui+1 , . . . , u2g−2 ) 7→ (u0 , u1 , . . . , ui ) and let ⊂ JP2g−2 K be the kernel, which has codimension i + 1. Then • one has a filtration KP : i KP

0 1 2g−3 2g−2 KP := JP2g−2 K ⊃ KP ⊃ . . . ⊃ KP ⊃ KP = (0).

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The space H 0 (K) can be identified with a g-dimensional subspace of JP2g−2 K via the map:

8 2g−2

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