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Geometry And Topology

Combinatorial and Geometric Structures and Their by A. Barlotti

By A. Barlotti

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We observe t h a t any plane o f AG(r,q) cannot meet K i n t h e same number o f points, say k ' . I n f a c t , i f i t were so, f o r a f i x e d m-secant l i n e t o f K, r-2 i as the number o f planes through i t i s 8r-2 = C q and every plane through i=o t i s k'-secant o f K, we should have k = (k'-m)e t m . S i m i l a r l y , f o r a nr-2 -secant l i n e we have k = ( k ' n)er-2+n; e q u a l i z i n g we have t h e absurd con- - e r-2 (n-m) dition = n - m . 3), planes . say klyk2 t h a t i s K i s k i n d (kly (kl < k2), k2) w i t h respect t o L e t 6 be a d i r e c t i o n o f AG(r,q).

The problem o f determining t h e k-sets o f k i n d (m,n) i n A(2,q) p r e l i m i n a r l y o f the determination o f the admissible p a i r s (m,n) i n v e s t i g a t i n g the r e a l existence o f the r e l a t e d k-sets. 17) i s admissible. 14) (where we p u t II i n A(2,q) h/2, / 2). 18) We v e r i f y t h a t f o r q < 2 f q ) / 2. 15) do n o t e x i s t . 9) and < R < h-2. 13), i f II = h - 1 and p = 2 we absurd 1 < s < 0. 13) i t i s 1 < II < h-2. 8), II h-1 - becomes c2 3qc t 2q(q t 1) = 0, which must have i n t e g e r roots, whence A = q(q-8) must be a square and i t happens o n l y when q = 9.

Napoli, 1973, 1-13. A. B a r l o t t i , Sui {k,nl-archi ( 3 ) 2 (1956) 553-556. T q' Relazio- d i un piano l i n e a r e f i n i t o , B o l l . , F. Buekenhout, Existence o f u n i t a l s i n f i n i t e t r a n s l a t i o n planes o f order q square w i t h a kernel o f order q, Geometriae Dedicata, 5 (1976) 189-194. V. Ceccherini, Su c e r t i {k,n)-archi d e d o t t i da curve piane e s u l l e ( r > 2), Rend. , (2)6(1969) { k , n l - c a l o t t e d i t i p 0 (0,n) d i un S r,q 185-1 95.

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