4.1 Plücker quadric and Gr(2,4). Let 푉 be 4-dimensional vector space. e set of all 2-di- mensional vector subspaces 푈 ⊂ 푉 is called the grassmannian Gr(2, 4) = Gr(2, 푉). More geomet- rically, the grassmannian Gr(2, 푉) is the set of all lines ℓ ⊂ ℙ = ℙ(푉). Sending 2-dimensional vector subspace 푈 ⊂ 푉 to 1-dimensional subspace 훬 푈 ⊂ 훬 푉 or, equivalently, sending a line (푎푏) ⊂ ℙ(푉) to 한 ⋅ 푎 ∧ 푏 ⊂ 훬 푉 , we get the Plücker embedding
픲 ∶ Gr(2, 4) ↪ ℙ = ℙ(훬 푉). (4-1)
Its image consists of all decomposable¹ grassmannian quadratic forms 휔 = 푎∧푏 , 푎, 푏 ∈ 푉. Clearly, any such a form has zero square: 휔 ∧ 휔 = 푎 ∧ 푏 ∧ 푎 ∧ 푏 = 0. Since an arbitrary form 휉 ∈ 훬 푉 can be wrien¹ in appropriate basis of 푉 either as 푒 ∧ 푒 or as 푒 ∧ 푒 + 푒 ∧ 푒 and in the laer case 휉 is not decomposable, because of 휉 ∧ 휉 = 2 푒 ∧ 푒 ∧ 푒 ∧ 푒 ≠ 0, we conclude that 휔 ∈ 훬 푉 is decomposable if an only if 휔 ∧ 휔 = 0. us, the image of (4-1) is the Plücker quadric
푃 ≝ { 휔 ∈ 훬 푉 | 휔 ∧ 휔 = 0 } (4-2)
If we choose a basis 푒 , 푒 , 푒 , 푒 ∈ 푉, the monomial basis 푒 = 푒 ∧ 푒 in 훬 푉, and write 푥 for the homogeneous coordinates along 푒 , then straightforward computation
푥 ⋅ 푒 ∧ 푒 ∧ 푥 ⋅ 푒 ∧ 푒 = 2 푥 푥 − 푥 푥 + 푥 푥 ⋅ 푒 ∧ 푒 ∧ 푒 ∧ 푒 < < implies that 푃 is given by the non-degenerated quadratic equation
푥 푥 = 푥 푥 + 푥 푥 .
Exercise 4.1. Check that the Plücker embedding (4-1) takes a subspace spanned by 푢 = ∑ 푢 푒 , 푤 = ∑ 푤 푒 to grassmannian quadratic form with coefficients 푥 = 푢 푤 − 푢 푤 , i.e. sends
푢 푢 푢 푢 푢 푢 a matrix to the collection of its six 2 × 2-minors 푥 = det . 푤 푤 푤 푤 푤 푤 In coordinate-free terms, 푃 = 푉(푞) for the canonical up to a scalar factor quadratic form 푞 on 훬 푉 defined by prescription
∀ 휔 , 휔 ∈ 훬 푉 휔 ∧ 휔 = 푞̃(휔 , 휔 ) ⋅ 훺 , (4-3)
where 훺 ∈ 훬 푉 ≃ 한 is any fixed non-zero vector (unique up to proportionality). Since 휔 ∧ 휔 = 휔 ∧ 휔 for even grassmannian polynomials, the form 푞̃(휔 , 휔 ) is symmetric. Lemma 4.1
ℓ ∩ ℓ ≠ ∅ in ℙ ⟺ 푞̃(픲(ℓ ), 픲(ℓ )) = 픲(ℓ ) ∧ 픲(ℓ ) = 0 in ℙ .
Proof. Let ℓ = ℙ(푈 ), ℓ = ℙ(푈 ). If 푈 ∩ 푈 = 0, then 푉 = 푈 ⊕ 푈 and we can choose a basis 푒 , 푒 , 푒 , 푒 ∈ 푉 such that ℓ = (푒 푒 ) , ℓ = (푒 푒 ) . en 픲(ℓ ) ∧ 픲(ℓ ) = 푒 ∧ 푒 ∧ 푒 ∧ 푒 ≠ 0 . If ℓ = (푎푏) and ℓ = (푎푐) are intersecting in 푎, then 픲(ℓ ) ∧ 픲(ℓ ) = 푎 ∧ 푏 ∧ 푎 ∧ 푐 = 0.
¹i.e. factorized as exterior product of two vectors ¹see example 3.5 on p. 60
73 74 §4 Grassmannians
Corollary 4.1 e Plücker embedding (4-1) is really injective and establishes a bijection between the grassman- nian Gr(2, 4) and the Plücker quadric (4-2).
Proof. For any two lines ℓ ≠ ℓ on ℙ there exists a third line ℓ which intersect ℓ and does not intersect ℓ . en 픲(ℓ ) ∧ 픲(ℓ) = 0 and 픲(ℓ ) ∧ 픲(ℓ) ≠ 0 imply 픲(ℓ ) ≠ 픲(ℓ ).
Corollary 4.2 For any point 푝 = 픲(ℓ) ∈ 푃 the intersection 푃 ∩ 푇 푃 consists of all 픲(ℓ ) such that ℓ ∩ ℓ ≠ ∅.
Proof. is follows from cor. 2.4 on p. 28 and lemma 4.1 above.
4.1.1 Line nets and line pencils in ℙ . A family of lines on ℙ is called a net of lines if the Plücker embedding sends it to some plane 휋 ⊂ 푃 ⊂ ℙ . Any plane 휋 ⊂ 푃 is spanned by a triple of non collinear points 푝 = 픲(ℓ ), 푖 = 1, 2, 3, and lies in the tangent space to each of them. us, 휋 ⊂ 푃 ∩ 푇 푃 ∩ 푇 푃 ∩ 푇 푃 . By lemma 4.1 and cor. 4.2, the corresponding net of lines consist of all lines that intersect 3 given pairwise intersecting lines ℓ ⊂ ℙ . ere are exactly two geometrically different possibilities fo that:
훼-net consists of all lines passing through a given point 푂 ∈ ℙ and corresponds to 훼-plane 휋 (푂) ⊂ 푃 spanned by Plücker's images of any 3 non-coplanar lines passing through 푂
훽-net consists of all lines in a given plane 훱 ∈ ℙ and corresponds to 훽-plane 휋 (훱) ⊂ 푃 spanned by Plücker's images of any 3 lines laying in 훱 without common intersection.
Any two planes of the same type are always intersecting in precisely one point:
휋 훱 ∩ 휋 훱 = 픲 훱 ∩ 훱 , 휋 푂 ∩ 휋 푂 = 픲 (푂 푂 ) .
Two planes of different types 휋 (훱) , 휋 (푂) either do not intersect each other (if 푂 ∉ 훱) or are intersecting along a line (if 푂 ∈ 훱) that corresponds to the pencil of lines ℓ ⊂ ℙ passing through 푂 and laying in 훱.
Exercise 4.2. Show that there are no other pencils of lines in ℙ , i.e. each line laying on 푃 ⊂ ℙ has a form 휋 (훱) ∩ 휋 (푂) for some 푂 ∈ 훱.
4.1.2 Cell decomposition. Fix some 푝 ∈ 푃 and a hyperplane 퐻 ≃ ℙ complementary to 푝 inside 푇 푃 ≃ ℙ . en intersection 퐶 = 푃 ∩ 푇 푃 is a simple cone with vertex at 푝 over a smooth quadric 퐺 = 퐻 ∩ 푃 that can be thought of as the Segre quadric in ℙ = 퐻. Choosing some point 푝 ∈ 퐺 and writing 휋 , 휋 for planes spanned by 푝 and two lines on 퐺 passing through 푝 , wee get a stratification of the Plücker quadric 푃 by closed subvarieties shown on fig. 4⋄1 :
휋 (4-4) ; o
- / / 푝 휋 ∩ 휋 q ? 퐶 푃
# / 휋 4.1. Plücker quadric and Gr(2, 4) 75
It decomposes 푃 as disjoint union of open cells² isomorphic to affine spaces:
⎛ 픸 ⎞ Gr(2, 4) = 픸 ⊔ 픸 ⊔ ⎜ ⊔ ⎟ ⊔ 픸 ⊔ 픸 ⎜ ⎟ ⎝ 픸 ⎠
πβ p′
πα
G ⊂ H
p ̸∈ H
H ≃ P3
Рис. 4⋄1. e cone 퐶 = 푃 ∩ 푇 푃 ⊂ ℙ = 푇 푃.
It starts with {푝} ≃ 픸 , then stays projective line without this point: 휋 ∩ 휋 ∖푝 ≃ 픸 , then the pair of projective spaces without this line: 휋 ∖ 휋 ∩ 휋 ≃ 휋 ∖ 휋 ∩ 휋 ≃ 픸 , then we have the cone 퐶 over 퐺 without these two planes: 퐶 ∖ 휋 ∪ 휋 ≃ 픸 × 퐺 ∖ 퐺 ∩ 푇 퐺 . Finally, there are natural identifications 퐺 ∖ 퐺 ∩ 푇 퐺 ≃ 픸 and 푄 ∖ 퐶 ≃ 픸 provided by the next lemma.
Lemma 4.2
Projection 푝 ∶ 푄 → 훱, of a smooth quadric 푄 ⊂ ℙ from a point 푝 ∈ 푄 onto hyperplane 훱 ∌ 푝, − establishes a bijection between 푄 ∖ 푇 푄 and 픸 = 훱 ∖ 푇 푄.
Proof. Each non-tangent line passing through 푝 does intersect 푄 in precisely one point³ different − from 푝. All these lines stay in bijections with the points of 픸 = 훱 ∖ 푇 푄.
²each affine cell of this decomposition is an open dense subset of the corresponding stratum in (4-4) complementary to the union of all the strata of lower dimension contained in the stratum we deal with ³if we write 푥 for this point and put 푦 = (푝푥) ∩ 훱, then it follows from Vieta formulas that 푥 and 푦 are rational functions of each other; thus, the bijection of lemma is actually an isomorphism of affine algebraic varieties 76 §4 Grassmannians
Exercise 4.3. If you have some experience in CW-topology, show that integer homologies of complex grassmannian are
⎧0 for odd 푚 and 푚 > 8 ⎪ 퐻 Gr(2, ℂ ), ℤ = ⎨ℤ for 푚 = 0, 2, 6, 8 ⎪ ⎩ℤ ⊕ ℤ for 푚 = 4
Try to compute integer homologies 퐻 Gr(2, ℝ ), ℤ of the real grassmannian (the bound- ary maps are non trivial here).
4.2 Lagrangian grassmannian LGr(2, 4) and lines on a smooth quadric in ℙ . Let us equip 4-dimensional vector space 푉 with non-degenerated skew-symmetric bilinear form 훺 and fix some non-zero vector 훿 ∈ 훬 푉. en there exists a unique grassmannian quadratic form 휔 ∈ 훬 푉 satisfying ∀ 푎, 푏 ∈ 푉 휔 ∧ 푎 ∧ 푏 = 훺(푎, 푏) ⋅ 훿 . (4-5)
Write 푊 = Ann 푞̂(휔) for orthogonal complement to 휔 w.r.t. the Plücker quadratic form 푞 on 훬 푉 defined in formula (4-3), p. 73. en 훧 = ℙ(푊) ≃ ℙ is the polar hyperplane of 휔 w.r.t. the Plücker quadric 푃 ∈ ℙ = ℙ(훬 푉). Exercise 4.4. Check that 휔 ∉ 푃.
Since 휔 ∉ 푃 the intersection 푅 ≝ 훧 ∩ 푃 is a smooth quadric in 훧 ≃ ℙ . Its points stay in bijection with lagrangian subspaces¹ in 푉 w.r.t. symplectic form 훺, because of (4-5) and 4-3 which say together that line (푎푏) ⊂ ℙ has 훺(푎, 푏) = 0 iff 푞̃(휔, 푎 ∧ 푏) = 0. By this reason, 푅 is called lagrangian grassmannian of symplectic form 훺 and is usually denoted by LGr(2, 4) = LGr(훺, 푉). It follows from general theory developed in n∘ 2.5 that 푅 does not contain planes but is filled by lines in such a way that lines passing through a given point 푟 ∈ 푅 rule a simple cone over a smooth conic with the vertex at 푟. e variety of lines laying in a given algebraic manifold 푋 is called the Fano Variety of 푋 and is denoted by 퐹(푋).
Proposition 4.1
ere is well defined isomorphism ℙ = ℙ(푉) ⥲ 퐹(푅) sending a point 푝 ∈ ℙ(푉) to the pencil of all lagrangian lines in ℙ(푉) passing through 푝.
Proof. We have seen in n∘ 4.1.1 (esp. exrs. 4.2 on p. 74) that each line 퐿 ⊂ 푃 is an intersection of
훼- and 훽-planes: 퐿 = 휋 ∩ 휋(훱), i.e. consists of all lines passing through some point 푝 ∈ ℙ(푉) and laying in some plane 훱. If 퐿 ⊂ 푅 = 푃 ∩ 훧, then 퐿 consists of all lagrangian lines passing trough 푝 and laying in 훱. On the other hand, a line (푝푥), which passes through a given point 푝 ∈ ℙ(푉), is lagrangian iff 훺(푝, 푥) = 0. us, all lagrangian lines passing through an arbitrary point 푝 lie a plane that is orthogonal to 푝 w.r.t. symplectic form 훺, that is, form a pencil.
¹푛-dimensional subspace 푈 in 2푛-dimensional subspace 푉 equipped with non-degenerated skew-sym- metric bilinear form 훺 is called lagrangian if 훺(푢, 푤) = 0 for all 푢, 푤 ∈ 푈; thus lagrangian subspaces are skew-symmetric analogues of maximal isotropic subspaces of non-degenerated symmetric forms 4.3. Grassmannians Gr(푘, 푛) 77
4.3 Grassmannians Gr(k,n). A variety of all of all 푚-dimensional vector subspaces in a given 푑-dimensional vector space 푉 is called a grassmannian variety and is denoted by Gr(푚, 푑) or by Gr(푚, 푉) , when the nature of 푉 is essential. In projective world, grassmannian Gr(푚, 푑) parametrizes (푚 − 1)-dimensional projective subspaces in ℙ − . Simplest grassmannians are the × ∗ projective spaces ℙ = ℙ(푉) = Gr(1, 푉) = Gr(1, 푛 + 1) and ℙ = ℙ(푉 ) = Gr(푛, 푉) = Gr(푛, 푛 + 1). More generally, the duality 푈 ↔ Ann 푈 establishes canonical bijection Gr(푚, 푉) ↔ Gr(푑 − 푚, 푉∗), where 푑 = dim 푉. Simplest grassmannian besides the projective spaces is Gr(2, 4) considered in the previous section. Exercise 4.5. Let dim 푉 = 4 and 푞̂ ∶ 푉 ⥲ 푉∗ be a correlation provided by some smooth quadric 푄 = 푉(푞) ⊂ ℙ(푉). Show that prescription 푈 ↦ Ann 푞̂(푈) defines an automorphism Gr(2, 푉) ⥲ Gr(2, 푉) that sends 훼-planes on Gr(2, 4) to the 훽-planes and vice versa. . 4.3.1 e Plücker embedding takes 푚-dimensional subspace 푈 ⊂ 푉 to the 1-dimensional subspace 훬 푈 ⊂ 훬 푉. is gives the mapping
픲 ∶ Gr(푚, 푉) ↪ ℙ(훬 푉). (4-6)
If vectors 푢 , 푢 ,…, 푢 form a basis in subspace 푈 ⊂ 푉, then 픲(푈) = 한 ⋅ 푢 ∧ 푢 ∧ ⋯ ∧ 푢 . Since for any two distinct 푚-dimensional subspaces 푈 ≠ 푊 we can choose a basis
푣 , 푣 ,…, 푣 , 푢 , 푢 ,…, 푢 − , 푤 , 푤 ,…, 푤 − , 푣 − , 푣 − + ,…, 푣 ∈ 푉 such that 푣 , 푣 ,…, 푣 form a basis of 푈 ∩ 푊 and 푢 , 푢 ,…, 푢 − , 푤 , 푤 ,…, 푤 − complete it to some bases of 푈 , 푊, the mapping (4-6) sends 푈 and 푉 to distinct basic monomials
푣 ∧ ⋯ ∧ 푣 ∧ 푢 ∧ ⋯ ∧ 푢 − ≠ 푣 ∧ ⋯ ∧ 푣 ∧ 푤 ∧ ⋯ ∧ 푤 − of 훬 푉. us, the Plücker embedding (4-6) is actually injective and establishes bijection between Gr(푚, 푑) and the variety of decomposable grassmannian polynomials 휔 ∈ 훬 푉. By prop. 3.4 on p. 69, the laer is described as an intersection of quadrics provided by the Plücker relations from the statement (3) of prop. 3.4. Algebraically, the grassmannian variety Gr(푚, 푑) ⊂ ℙ(훬 푉) is a straightforward skew-com- mutative analogue of the Veronese variety 푉(푚, 푑) ⊂ ℙ(푆 푉) : both consist of non-zero homo- geneous degree 푚 polynomials of the maximal degeneracy, that is, having the minimal possi- ble non-zero linear support. For ordinary commutative polynomial 푓 ∈ 푆 푉 this means that dim Supp 푓 = 1 and 푓 = 푣 for some 푣 ∈ 푉. For non-zero grassmannian polynomial 휔 ∈ 훬 푉 the minimal dim Supp 휔 = 푚 and in this case 휔 = 푤 ∧푤 ∧ ⋯ ∧푤 for some 푤 , 푤 ,…, 푤 ∈ 푉.
4.3.2 Matrix notations and the Plücker coordinates. As soon a basis 푒 , 푒 ,…, 푒 ∈ 푉 is chosen, one can represent a point 푈 ∈ Gr(푚, 푑) by an equivalence class of 푚 × 푑-matrix 퐴(푈) whose rows are the coordinates of vectors 푢 , 푢 ,…, 푢 ⊂ 푈 that form a basis in 푈. Another choice of basis in 푈 changes 퐴(푈) via le multiplication by a matrix 퐶 ∈ GL (한). us, the points of grassmannian Gr(푚, 푑) are the le GL (한)-orbits in Mat × (한). ese agrees with homogeneous coordinates on ℙ(푉) = Gr(1, 푉), which are the rows (i.e. 1 × 푑-matrices) up to ∗ multiplication by elements of GL (한) = 한 . Choosing the standard monomial basis 푒 = 푒 ∧ 푒 ∧ ⋯ ∧ 푒 in 훬 푉, one can describe the Plücker embedding (4-6) as a map that takes matrix 퐴(푈) to a point whose coordinate 푥 in the 78 §4 Grassmannians
basis 푒 equals the maximal minor 푎 (푈) of 퐴(푈) situated in columns 푖 , 푖 ,…, 푖 . Indeed,
푢 ∧ 푢 ∧ ⋯ ∧ 푢 = 푎 푒 ∧ 푎 푒 ∧ ⋯ ∧ 푎 푒 = = sgn(휎)훼 푎 ⋯ 푎 푒 ∧ 푒 ∧ ⋯ ∧ 푒 = 푎 푒 ( ) ( ) ( ) ⩽ < <⋯< ⩽ ∈픖 where 푎 = det 푎 . Maximal minors 푎 = 푎 (푈) of 퐴(푈) are called the Plücker coordinates of , 푈. It follows from example 3.3 that the le multiplication 퐴(푈) ↦ 퐶 ⋅ 퐴(푈) by some 퐶 ∈ GL (한) multiplies all the Plücker coordinates 푎 (푈) by det 퐶. 4.3.3 Affine charts and affine coordinates. e standard affine cards 푈 ⊂ ℙ(훬 푉), where 퐼 runs through increasing collections 푖 , 푖 ,…, 푖 , cover the grassmannian Gr(푚, 푉). Let us write 풰 ⊂ Gr(푚, 푉) for the intersection 푈 ∩ Gr(푚, 푉) and call it a standard affine chart on Gr(푚, 푉). is chart consists of all 푈 ⊂ 푉 such that 퐴(푈) has non zero maximal minor in the columns 푖 , 푖 ,…, 푖 . Geometrically, this means that 푈 ⊂ 푉 is isomorphically mapped onto coordinate subspace of 푉 spanned by the basic vectors 푒 with 푖 ∈ 퐼 by the projection along the complementary coordinate subspace spanned by 푒 with 푗 ∉ 퐼. In particular, there exist a unique basis of 푈 that consists of the preimages of basic vectors 푒 , 푖 ∈ 퐼 , under this projection. ( ) ( ) Algebraically, this means that each 푈 ∈ 풰 has a unique matrix representation 퐴 = 퐴 (푈) ( ) ( ) such that 푚 × 푚-submatrix 퐴 ⊂ 퐴 situated in 퐼-columns is the identity 푚 × 푚 matrix. is ( ) − matrix representation is obtained from an arbitrary representation 퐴 = 퐴(푈) as 퐴 = 퐴 ⋅ 퐴 , where 퐴 ⊂ 퐴 is 푚 × 푚-submatrix formed by 퐼-columns. us, the points of the standard chart 픘 ⊂ Gr(푚, 푑) stay in bijection with 푚 × 푑 matrices 퐴( ) with 퐸 in 퐼-columns and can be identified with the affine space 픸 ( − ) coordinated by the ( ) ( ) matrix elements 푎 of 퐴 staying outside the columns (푖 , 푖 ,…, 푖 ). In particular, dim Gr(푚, 푑) = 푚(푑 − 푚).
Exercise 4.6. If you had deal with differential, analytic, or algebraic geometry, check that real (resp. complex, or arbitrary) grassmannians are smooth (resp. holomorphic, or algebraic) manifolds.
4.4 Cell decomposition. e Gauss method shows that each 푈 ⊂ 푉 admits a unique basis
푢 , 푢 ,…, 푢 ∈ 푈 whose matrix 퐴 is a reduced step matrix. By the definition, this means that there is the identity 푚 × 푚-submatrix 퐸 ⊂ 퐴 situated in some columns 푗 , 푗 ,…, 푗 and each row of 퐴 vanishes at the le of the unity coming from this identity submatrix, that is, for each
푖 = 1, 2,…, 푚 and any 푗 < 푗 푎 = 0. Exercise 4.7. Make it sure that the rows of distinct reduced step matrices span distinct sub- spaces in 한 . us, there exist a bijection between Gr(푚, 푑) and strong step (푚 × 푑)-matrices of rank 푚. e laer split into disjoint union of affine spaces. Namely, all strong step matrices of prescribed shape, i.e. containing the identity submatrix in prescribed columns 푖 , 푖 ,…, 푖 , have exactly
푚푑 − 푚 − (푖 − 1) − (푖 − 2) − ⋯ − (푖 − 푚) = dim Gr(푚, 푑) − (푖 − 휈) = 4.4. Cell decomposition 79 free entries to put there any constants from 한. Hence, Gr(푚, 푑) is a disjoint union of affine cells 프 enumerated by increasing collections 퐼 = (푖 , 푖 ,…, 푖 ) ⊂ (1, 2,…, 푑). e 퐼-th cell 프 is isomorphic to affine space and has codimension ∑ (푖 − 휈) = |퐼| − ( + ) in Gr(푚, 푑). = 4.4.1 Young diagram notations. Another common way of numbering the disjoint affine cells 프 ⊂ Gr(푚, 푑) replaces increasing subset 퐼 = 푖 , 푖 ,…, 푖 by a partition 휆 = (휆 , 휆 ,…, 휆 ) that is non-decreasing collection of non-negative integers
휆 ⩾ 휆 ⩾ ⋯ ⩾ 휆 ⩾ 0 whose 휈 th from the right element 휆 + − is equal to 휈 th from the le difference (푖 − 휈) in the increasing collection 푖 , 푖 ,…, 푖 . us, the identity 푚 × 푚-submatrix of reduced step matrix of type 휆 = (휆 , 휆 ,…, 휆 ) is situated in columns 푖 = (푚 + 1 − 휈) + 휆 + − . In other words, 휆 equals the difference that appears in the 휈 th row from the boom between the actual position of the lemost non-zero element and the lemost possible position for such element. Partitions are visualised by means of Young diagrams, that is, aligned to the le cellular strips of lengths 휆 , 휆 ,…, 휆 . Total number of cells |휆| ≝ ∑ 휆 is called a weight² of the diagram. A number of rows ℓ(휆) ≝ max(푘 | 휆 > 0) is called a length of the diagram. For example, partition
(4, 4, 2, 1) ↭ has length 4 and weight 11. In grassmannian Gr(4, 10) it defines 13-dimensional affine cell formed by the subspaces representable by reduced step matrices of shape
⎛0 1 ∗ 0 ∗ ∗ 0 0 ∗ ∗⎞ ⎜0 0 0 1 ∗ ∗ 0 0 ∗ ∗⎟ . ⎜0 0 0 0 0 0 1 0 ∗ ∗⎟ ⎜ ⎟ ⎝0 0 0 0 0 0 0 1 ∗ ∗⎠ e differences between actual lemost units (colored in blue) and the lemost possible positions for them (colored in red) listen from the boom to the top are 4, 4, 2, 1, i.e. coincide with the partition. e zero partition (0, 0, 0, 0) corresponds to the lemost possible positions of steps and de- fines 24-dimensional affine cell of spaces representable by matrices of the shape
⎛1 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗⎞ ⎜0 1 0 0 ∗ ∗ ∗ ∗ ∗ ∗⎟ ⎜0 0 1 0 ∗ ∗ ∗ ∗ ∗ ∗⎟ ⎜ ⎟ ⎝0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗⎠ i.e. the standard affine chart 픘( , , , ) of the grassmannian Gr(4, 10). Maximal possible partition (6, 6, 6, 6) whose Young diagram exhausted the whole of rectangle
²Young diagrams of a given weight 푛 stay in bijection with partitions of 푛 into a sum of non-ordered positive integers (this explains the terminology) 80 §4 Grassmannians corresponds to the zero-dimensional one point cell represented by the reduced step matrix of the rightmost possible shape ⎛0 0 0 0 0 0 1 0 0 0⎞ ⎜0 0 0 0 0 0 0 1 0 0⎟ ⎜0 0 0 0 0 0 0 0 1 0⎟ ⎜ ⎟ ⎝0 0 0 0 0 0 0 0 0 1⎠ Exercise 4.8. Make it sure that collections of increasing indexes 퐼 ⊂ (1, 2,…, 푑) stay in bijec- tions with Young diagrams 휆 contained in rectangle of size 푚 × (푑 − 푚). ∘ us, grassmannian Gr(푚, 푑) = ⨆ 휎 is disjoint union of affine spaces
∘ ( − )−| | 휎 = 픸 called open Schubert cells and numbered by the Young diagrams 휆 contained in the rectangle of ∘ size 푚 × (푑 − 푚). e closure of 휎 in Gr(푚, 푑) is called (closed) Schubert cycle. Example 4.1 (homologies of complex grassmannians)
Closed Shubert cycles 휎 form a free basis of abelian group of integer homologies of complex grassmannian 훬(푚, 푑) ≝ 퐻∗ Gr(푚, ℂ ), ℤ , because the cell decomposition just constructed does not contain cells of odd real dimension. e laer means that all boundary operators in the cell chain complex vanish.
∗ ∘ ∘ Exercise 4.9 . List all open cells 휎 containing in the closure of a given open cell 휎 and try to evaluate the boundary operators in the cell chain complex of real grassmannian Gr 푚, ℝ . 4.4.2 Scubert calculus. Topological intersection of cycles provides abelian group 훬(푚, 푑) with a structure of commutative ring. It turns to be a truncated ring of symmetric functions:
훬(푚, 푑) ≃ ℤ[휀 , 휀 ,…, 휀 ]/(휂 − + ,…, 휂 − , 휂 ) (4-7) where 휀 and 휂 stay for 푘 th elementary¹ and 푘 th complete² symmetric polynomials in 푥 , 푥 ,…, 푥 . By the main theorem about symmetric functions, all 휂 are uniquely expanded as polynomials in 휀 , 휀 ,…, 휀 and factorisation in 4-7 is modulo the ideal spanned by those polynomials expand- ing 휂 − + ,…, 휂 trough 휀 's. e isomorphism (4-7) sends a Schubert cycle 휎 to the Schur polynomial 푠 , that is the sum of all monomials in 푥 , 푥 ,…, 푥 obtained as follows: fill the cells of diagram 휆 by leers 푥 , 푥 ,…, 푥 (each leer can be used any number of times) in such a way that the indexes of variables strictly increase along the columns from the top to the boom and non-strictly increase along the rows from the le to the right, then multiply all the entries into a monomial of degree 휆 푠 휂 푘 푘 푠 휀 1 | |. For example, ( ) = , where ( ) means one row of length , and = , where = (1,…, 1) means one column of height 푘. All proofs of the isomorphism (4-7) that I know are non direct. Anyway, they use quite sophisticated combinatorics of symmetric functions besides the geometry of grassmannians. e geometric part of the proof establishes two basic intersection rules:
¹sum of all 푘-linear monomials of degree 푘 ²sum of all degree 푘 monomials in 푥 , 푥 ,…, 푥 at all 4.4. Cell decomposition 81
1) An intersection of cycles 휎 , 휎 of complementary codimensions |휆| + |휇| = 푚(푑 + 푚) is non empty iff their diagrams are complementary³ and in this case the intersection consists of one point corresponding to the matrix of shape 휆 whose ∗-entries equal zero.
휆 휎 휎 휎 휎 휎 휎 휇 휈 2) e Pieri rules: for any diagram we have ( ) = ∑ and = ∑ , where , run through Young diagrams obtained by adding 푘 cells to 휆 in such a way that all added cells stay in distinct rows in 휇 and stay in distinct columns in 휈.
e details can be found in P. Griffits, J. Harris. Principles of Algebraic Geometry, I. Combinatorial part of the proof verifies that the Schur polynomials satisfy the Pieri rules and form a basis of the ℤ-module 훬 of symmetric polynomials in countable number of variables with integer coefficients. Direct computation shows that the Pieri rules uniquely determine the multiplicative structure on 훬. is leads to surjective homomorphism 훬 ↠ 훬(푚, 푑) sending 푠 ↦ 휎 . Its kernel is uniquely determined as soon we know that the classes of 푠 with 휆 ⊆ (푑 ) form a basis in the factor ring and satisfy there the first intersection rule mentioned above. e details can be found in two remarkable W. Fulton's books Young Tableaux and Intersection eory. Avoiding general theory, we compute the intersection ring of Gr(2, 4) in example 4.2. Exercise 4.10. Verify that in the notations of n∘ 4.1.1 six Schubert cycles on the Plücker quadric
푃 = Gr(2, 4) ⊂ ℙ are: 휎 = 푃; 휎 = 푝 = (0 ∶ 0 ∶ 0 ∶ 0 ∶ 0 ∶ 1) ∈ ℙ ; 휎 = 푃 ∩ 푇 푃; 휎 = 휋 (푂), where 푂 = (0 ∶ 0 ∶ 0 ∶ 1) ∈ ℙ ; 휎 = 휋 (훱), where 훱 ⊂ ℙ is given by linear equation 푥 = 0; 휎 = 휋 (푂) ∩ 휋 (훱).
Example 4.2 (intersection theory on Gr(2, 4))
If codim휎 + codim휎 < 4, then 휎 휎 = ∅, certainly. Intersections of cycles of complementary ∘ codimensions was described actually in n 4.1.1 and exrs. 4.2: 휎 휎 = 휎 = 휎 = 휎 and 휎 휎 = 0. By the same geometric reasons 휎 휎 = 휎 휎 = 휎 . In order to compute 휎 , let us realise 휎 as 휎 (ℓ) = 푃 ∩푇픲(ℓ)푃 = {ℓ ⊂ ℙ | ℓ∩ℓ ≠ ∅} . en 휎 is a homology class of an intersection 휎 (ℓ)∩휎 (ℓ ) that generically is represented by the Segre quadric shown in fig. 4⋄1 on p. 75. Let us move ℓ to a position where ℓ ∩ ℓ ≠ ∅ but ℓ ≠ ℓ. Under this moving the Segre quadric is deformed inside its homology class to the union of two planes: 훼-net with the centre at 푂 = ℓ∩ℓ and 훽-net in the plane 훱 spanned by ℓ∪ℓ , i.e. we get 휎 (ℓ)∩휎 (ℓ ) = 휋 (푂)∪휋 (훱) . us, 휎 = 휎 + 휎 . is leads to «topological» solution of exrs. 2.10 and prb. 2.10 asked how many lines does intersect 4 given mutually skew lines in ℙ . If 4 given lines are general enough⁴ , then formal computation in 훬(2, 4) : 휎 = (휎 + 휎 ) = 휎 + 휎 = 2휎 tells us that there are 2 such lines in general.
³i.e. can be fied together without holes and overlaps to assemble 푚 × (푑 − 푚) rectangle ⁴namely, the intersection of cycles 휎 (ℓ ) provided by these lines represents the topological 4-fold self-intersection휎 82 Home task problems to §4
Home task problems to §4
Problem 4.1. Is there a complex 2 × 4-matrix whose set of 2 × 2-minors is a) {2, 3, 4, 5, 6, 7} b) {3, 4, 5, 6, 7, 8} ? If so, give an explicit example of such a matrix. Problem 4.2. Let 퐺 = 푉(푔) ⊂ ℙ = ℙ(푉) be a smooth quadric. Define a bilinear form 훬 푔 푔 (푣 , 푤 ) 푔 (푣 , 푤 ) on 훬 푉 by prescription 훬 푔 ( 푣 ∧ 푣 , 푤 ∧ 푤 ) ≝ det . Check that it 푔 (푣 , 푤 ) 푔 (푣 , 푤 ) is symmetric and non-degenerated and write down its Gram matrix in a standard monomial basis of 훬 푉 built of some 푔-orthonormal basis in 푉. Problem 4.3. In continuation of prb. 4.2 take 푔(퐴) = det 퐴 as the quadratic form on the space 푉 = Hom(푈−, 푈+) , where 푈± ≃ ℂ , write 훬 푔 for the smooth quadratic form on 훬 푉 that 푔 (푣 , 푣 ) 푔 (푣 , 푣 ) takes 푣 ∧ 푣 to the Gram determinant 훬 푔 푣 ∧ 푣 ≝ det , and write 푔 (푣 , 푣 ) 푔 (푣 , 푣 ) 푃 = {휔 ∈ 훬 푉 | 휔 ∧ 휔 = 0} ⊂ ℙ = ℙ(훬 푉) for the Plücker quadric. Show that a) the intersection of quadrics 푉(훬 푔) ∩ 푃 ⊂ ℙ consists of all lines in ℙ = ℙ(푉) tangent to the Segre quadric 퐺 = 푉(푔) ⊂ ℙ . b) the Plücker embedding Gr(2, 푉) ⥲ 푃 ⊂ ℙ(훬 푉) sends two line rulings of the Segre quadric
퐺 to a pair of distinct smooth conics 퐶± ⊂ 푃 that are cut out of the Plücker quadric by a pair of ∗ ∗ complementary planes 훬− = ℙ 푆 푈− ⊗ 훬 푈+ and 훬+ = ℙ 훬 푈− ⊗ 푆 푈+ embedded into ℙ = ℙ(훬 Hom(푈−, 푈+)) by means of decomposition (3-55) from prb. 3.6 on p. 71. ∗ ∗ c) both conics 퐶− ⊂ ℙ 푆 푈− ⊗ 훬 푈+ and 퐶+ ⊂ ℙ 훬 푈− ⊗ 푆 푈+ are the images of the ∗ ∗ Veronese embeddings ℙ(푈−) ⊂ ℙ 푆 푈− and ℙ(푈+) ⊂ ℙ 푆 푈+ , i.e. we have the following commutative diagram of the Plücker – Segre – Veronese interactions¹:
/ ℙ(푈 ) ℙ(푆 푈 ) ≃ 훬 O + Veronese +_ +
+ 훬 푈∗ ⊗ 푆 푈 Segre ⎛ − +⎞ + − / Plücker / ⎜ ⎟ ℙ × ℙ ∼ 퐺 ⊂ ℙHom(푈−, 푈+) 푃 ⊂ ℙ ⊕ ⎜ ∗ ⎟ ⎝푆 푈O − ⊗ 훬 푈+⎠ − ? ∗ Veronese / ∗ ℙ(푈−) ℙ(푆 푈−) ≃ 훬−
d) (Hodge's star) Associated with smooth quadretic form 푔 on 푉 is the Hodge star-operator
↦ ∗ ∗ ∶ 훬 푉 −−−−−−→ 훬 푉 ,
∗ defined by prescription ∀ 휔 , 휔 ∈ 훬 푉 휔 ∧ 휔 = 훬 푔 (휔 , 휔 ) ⋅ 푒 ∧ 푒 ∧ 푒 ∧ 푒 , where 푒 , 푒 , 푒 , 푒 ∈ 푉 is an orthonormal basis for 푔. Verify that this definition does not depend on a choice of orthonormal basis, find eigenvalues and eigenspaces of ∗, and show their place in the previous picture.
¹Plücker is dashed, because it takes lines to points Hints and answers for some exersices
∘ Exrs. 4.2. (Comp. with general theory from n 2.5 on p. 39.) e cone 퐶 = 푃 ∩ 푇 푃 consist of all lines passing through 푝 and laying on 푃. On the other hand, it consists of all lines joining its vertex 푝 with a smooth quadric 퐺 = 퐶 ∩ 퐻 cut out of 퐶 by any 3-dimensional hyperplane
퐻 ⊂ 푇 푃 complementary to 푝 inside 푇 푃 ≃ ℙ . us, any line on 푃 passing through 푝 has a form (푝푝 ) = 휋 ∩ 휋 , where 푝 ∈ 퐺 and 휋 , 휋 are two planes spanned by 푝 and two lines laying on the Segre quadric 퐺 and passing through 푝 (see fig. 4⋄1 on p. 75).
Exrs. 4.4. If 휔 ∈ 푃, then 훧 = 푇 푃 and 휔 = 픲(ℓ) for some lagrangian² line ℓ ⊂ ℙ(푉). is means that all lines in ℙ intersecting the lagrangian line ℓ are lagrangian too and contradicts with non- degeneracy of 훺.
²a line (푎푏) ⊂ ℙ(푉) is called lagrangian if 훺(푢, 푤) = 0 for all 푢, 푤 ∈ ℓ; this is equivalent to 훺(푎, 푏) = 0 and, by (4-5) and 4-3, to 푞̂ 휔, 픲(ℓ) = 0
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