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Get q-Clan Geometries in Characteristic 2 (Frontiers in PDF

By Ilaria Cardinali

ISBN-10: 3764385073

ISBN-13: 9783764385071

A q-clan with q an influence of two is such as a definite generalized quadrangle with a kin of subquadrangles every one linked to an oval within the Desarguesian aircraft of order 2. it's also corresponding to a flock of a quadratic cone, and as a result to a line-spread of three-d projective area and hence to a translation aircraft, and extra. those geometric gadgets are tied jointly via the so-called primary Theorem of q-Clan Geometry. The e-book supplies a complete evidence of this theorem, through a close research of the recognized examples. The collineation teams of the linked generalized quadrangles and the stabilizers in their linked ovals are labored out thoroughly.

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Ilaria Cardinali's q-Clan Geometries in Characteristic 2 (Frontiers in PDF

A q-clan with q an influence of two is similar to a definite generalized quadrangle with a relatives of subquadrangles each one linked to an oval within the Desarguesian aircraft of order 2. it's also akin to a flock of a quadratic cone, and therefore to a line-spread of three-d projective area and therefore to a translation aircraft, and extra.

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Additional info for q-Clan Geometries in Characteristic 2 (Frontiers in Mathematics)

Example text

53) 36 Chapter 2. The Fundamental Theorem For such a θ1 we can define the action of Tθ1 on the planes of the flock C as follows, where B1−1 = ac11 db11 : ⎡ ⎤ ⎡ ⎤ ⎛ xtπ λ1 a21 xt ⎢ yt ⎥ ⎢ ytπ ⎥ ⎜ 0 ⎥ ⎢ ⎥ ⎜ T θ1 : ⎢ ⎣ zt ⎦ → ⎣ ztπ ⎦=⎝ λ1 c21 1 1 0 λ1 a1 b1 λ1 b21 λ1 (a1 d1 + b1 c1 ) 0 λ1 c1 d1 λ1 d21 0 0 ⎞ ⎡ σ1 ⎤ x0π1 xt ⎢ σ1 ⎥ y0π1 ⎟ ⎟ ⎢ ytσ1 ⎥ . 54) Keep in mind that Eq. 52) is really just an efficient encoding of the description of Tθ1 given in Eq. 54). Suppose we have a second member θ2 of H with λ2 , σ2 , etc.

23) 26 Chapter 2. 1 (The Fundamental Theorem). Let C = {At ≡ C = {At ≡ so that A0 = xt 0 ( 00 yt zt 0) 0 xt yt 0 zt : t ∈ F } and : t ∈ F } be two (not necessarily distinct) q-clans normalized = A0 . Then the following are equivalent: (i) C ∼ C . (ii) The flocks F (C) and F (C) are projectively equivalent. (iii) GQ(C) and GQ(C) are isomorphic by an isomorphism mapping (∞) → (∞), [A(∞)] → [A (∞)], and (0, 0, 0) → (0, 0, 0). (iv) The associated spreads S(C) and S(C) are equivalent by a semilinear transformation leaving L∞ fixed and mapping the special reguli of S(C) to the special reguli of S(C) .

For now, we note that θ(id, I ⊗ aI) : (α, β, c) → (aα, aβ, a2 c) is an automorphism of G⊗ that leaves invariant each A(t) for t ∈ F˜ no matter what q-clan C is used to obtain A(t). In the present context Eq. 25) takes the following form: θ(σ1 , A1 ⊗ B1 ) ◦ θ(σ2 , A2 ⊗ B2 ) = θ(σ1 ◦ σ2 , Aσ1 2 A2 ⊗ B1σ2 B2 ). 38) and θ(σ, A ⊗ B) · θ(id, I ⊗ aI) · θ(σ, A ⊗ B)−1 = θ(id, I ⊗ aσ −1 I). 8. θ(σ, A ⊗ B) is linear provided σ = id. 9. θ(σ, A ⊗ B) is called special provided ∆ = det(B) = 1. Note that {θ(σ, I ⊗ I) : σ ∈ Aut(F )} ∼ = Aut(F ).

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q-Clan Geometries in Characteristic 2 (Frontiers in Mathematics) by Ilaria Cardinali


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