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§4 Grassmannians 73 §4 Grassmannians 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 훬푈 ⊂ 훬푉 or, equivalently, sending a line (푎푏) ⊂ ℙ(푉) to 한 ⋅ 푎 ∧ 푏 ⊂ 훬푉 , 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 ∈ 훬푉 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 ∈ 훬푉 | 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 푥푥 = 푥푥 + 푥푥 . Exercise 4.1. Check that the Plücker embedding (4-1) takes a subspace spanned by 푢 = ∑ 푢푒, 푤 = ∑ 푤푒 to grassmannian quadratic form with coefficients 푥 = 푢푤 − 푢푤, 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 훬푉 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 = (푒푒) , ℓ = (푒푒) . 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 ∩ 푇푃 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 (푂푂) . 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 푇푃 ≃ ℙ. en intersection 퐶 = 푃 ∩ 푇푃 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 p0 πα G ⊂ H p 62 H H ' P3 Рис. 4⋄1. e cone 퐶 = 푃 ∩ 푇푃 ⊂ ℙ = 푇푃. 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 ∖ 푇푄 and 픸 = 훱 ∖ 푇푄. 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 ∖ 푇푄. 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 ∈ 훬푉. en there exists a unique grassmannian quadratic form 휔 ∈ 훬푉 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.
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