Seminar on Differential Equations and Dynamical Systems by G. S. Jones

By G. S. Jones

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F, . , n ) . Baer cones a r e are m u t u a l l y P EXAMPLE. 1 and 3 . 3 imply f o r i n s t a n c e q- G t l Inp ( 2 , 4 , q ) 5 q2 -q S q t q - S q + l , i f t h e p r i m e power q i s a s q u a r e . Now we c o n s i d e r t h e a f f i n e _c_a_s~. 4 THEOREM. n , ( t , r , q ) s q r [ ( t + i ) q r - t - t PROOF. Any c o m p o n e n t B, o f a n n - f o l d ( w i t h n t 2 ) i s a t - b l o c k i n g s e t o f A. I-'. t-blocking set of A=AG(r,q) S o , by [ 2 1 , Cor 2 . 2 3 , w e h a v e IBiI 2 ( t t l ) q r - t - t , Hence, t h e assertion follows.

If G 2 ( h ) f 0, then L is not incident with the point of 4 ( M I . (ii) NO clique of G ( M ) contains L, I G , ( M ) I = 1 an& L is incident with the point 4 (&). (b) Any two cliques of Gl(3) are disjoint. PROOF. (a) We may suppose L E M . Using the notation of Lemma 3 we get t € I0,l) by Lemma 3 . Moreover, Lemma 3 implies that (1) (or (ii)) occurs if and only if t = 0 (or t = 1, respectively). 0 02 LEMMA 6. Let M1 and M2 4(A1) = 5 ( A 2 ) (a) be two distinct normal cliques. Then A 1 n M2 = 0 0.

In view of Lemma 4 we may assume without loss of generality that 5 ( 4 . ) with points p1 and p Lemma 5(a) sa s that a line L2 it incidknt with p. if and only ig no clique of contains L. Since G1(Mlt = T1(fi2), this shows that a line is incident with p1 if and only if it is incident with p2. Since r 1. 2, we Pi have 5,(fll) = 5 , ( M 2 ) . (b) In view of (a), one direction is obvious, Let us suppose A 1 n M 2 f 0. Define T = r M I ki € ,E, M nMl = 0 fl n M 2 } . G ( M 2 ) , we have Since M1, M 2 [ 5(M1) .

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