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Blocking Sets in the Projective Plane of Order Four
Annals of Discrete Mathematics 37 (1988) 43-50 0 Elsevier Science Publishers B.V. (North-Holland)
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BLOCKING SETS I N THE PROJECTIVE PLANE OF ORDER FOUR L u i g i a BERARDI and F r a n c o EUGENI Dipartimento d i Ingegneria E l e t t r i c a . U n i v e r s i t a ' d e L'Aquila ( I t a l i a ) I s t i t u t o d i Scienze. Universita' d i C h i e t i (Italia) We g i v e a c o m p l e t e c h a r a c t e r i z a t i o n o f b l o c k i n g s e t s i n PG(2,4).
1. INTRODUCTION A blocking set i n a f i n i t e i n c i d e n c e s t r u c t u r e ( P * 9 ,I) is a s u b s e t S o f P s u c h t h a t e v e r y b l o c k o f $ i s i n c i d e n t t o S and t o P-S a t l e a s t i n o n e p o i n t . The d e f i n i t i o n i m p l i e s t h a t i f S i s a b l o c k i n g s e t , t h e n P-S i s a b l o c k i n g s e t t o o . A b l o c k i n g s e t i s s a i d t o b e i r r e d u c i b l e i f v x E S t h e s e t S- ( x } i s n o t a blocking set. The c o n c e p t o f b l o c k i n s e t was i n t r o d u c e d , f o l l o w i n g a n i d e a o f Von Newmann and M o r g e n s t e r n (L205 , f o o t o n o t e 3, p. 4 6 9 ) , w i t h t h e name b l o c k i n g coal i t i o n s by M.Richardson i n 1956 [16], i n r e l a t i o n t o t h e s o - c a l l e d finite p r o j e c t i v e games (see a l s o [12]). I n 1966 J . D i P a o l a wrote two p a p e r s on t h e s e q u e s t i o n s w i t h t h e f i r s t r e s u l t s - i g p r o j e c t i v e p l a n e s of small o r d e r and i n t h e d u a l o f a S t e i n e r s y s t e m , [8J, L9J. I n 1970 A.Bruen i n t r o d u c e d t h e word " b l o c k i n g set" and began a sistematic s t u d y . S i n c e t h e n s e v e r a l a u t h o r s h a v e c o n t r i b u t e d t o t h i s s t u d y . R e c e n t l y i t h a s b e e n proved t h a t t h e e x i s t e n c e of b l o c k i n g sets i n p r o j e c t i v e o r a f f i n e s p a c e s o f h i g h d i m e n s i o n whoFe o r d e r i s l a r g e enough ( c f . [3], 171 , [14]).A u s e f u l l i s t c a n b e found i n [ 1 9 J . ~ erecall:
1.1 RESULT ( B r u e n [4], [S]). L e t S b e a b l o c k i n g set i n a p r o j e c t i v e p l a n e of o r d e r q.Then: q+Sq+l I S I q2-&j. Moreover, e q u a l i t y on t h e l e f t h a n d - s i d e h o l d s i f a n d o n l y i f S i s a B a e r s u b p l a n e . (On t h e r i g h t h a n d - s i d e i f f S is t h e complement of a B a e r s u b p l a n e ) . 1 . 2 RESULT (Bruen-Thas [6] , T a l l i n i [17] , [19] ). L e t S a n i r r e d u c i b l e s e t i n a p r o j e c t i v e p l a l i e o f o r d e r q . Then S I q S q + l , moreover t h e h o l d s i f and o n l y i f S i s a h e r m i t i a n arc.
blocking equality
S u p p o s e q = p h , p a p r i m e , q > 2 . We d e f i n e t h e f u n c t i o n m(q) as f o l l o w s : m ( q ) = G , i f q i s a s q u a r e ; m ( q ) = ( q + l ) / 2 , i f q i s a p r i m e and m(q)=ph-d, i f h > l is odd and where d d e n o t e s t h e g r e a t e s t d i v i s o r of h , d i f f e r e n t f r o m h. Then
1.3 RESULT ( B e r a r d i - E u g e n i [ I]). F o r any i n t e g e r k w i t h q+m(q)+l I k 5 q2 - m(q), t h e r e e x i s t s a b l o c k i n g s e t i n PG(2,q) h a v i n g e x a c t l y k e l e m e n t s . Many i n f o r m a t i o n s a b o u t sets o f PG(2,q). q e v e n , . i n t e r s e c t e d by a l i n e 0, q / 2 - 1, q / 2 , q/2+1 o r q p o i n t s , c a n b e found i n L l O ] .
in
The a i m o f t h i s p a p e r i s t o s t u d y t h e b l o c k i n g s e t s i n PG(2,4). Why j u s t t h i s c a s e ? We h a v e s e v e r a l r e a s o n s . a ) I f q=2, t h e r e i s no b l o c k i n g set i n PG(2,2); i f q = 3 w e h a v e i n PG(2.3) So q=4 i s r e a l l y t h e o n l y two b l o c k i n g s e t s up t o isomorphism (see [ 173). f i r s t case; moreover f o r a n y k w i t h 7 I k 1 1 4 w e h a v e a t least o n e b l o c k i n g set w i t h k p o i n t s , by 1.3. b) The s p a c e PG(3,4) i s t h e o n l y 3 - d i m e n s i o n a l case i n which t h e e x i s t e n c e of b l o c k i n g s e t s i s a n open problem. W e h o p e t h a t t h e s t u d y o f b l o c k i n g sets of PG(3,4) c a n depend on t h e i r s e c t i o n s w i t h a p l a n e t h a t are t h e b l o c k i n g sets o f FG(2,4). C ) I n [ 2 ] and [24] a l l b l o c k i n g sets i n t h e l i t t l e M a t h i e u s y s t e m s
L. Berwdi and F. Eugeni
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S ( 5 , 6 , 1 2 ) and i t s c o n t r a c t i o n s have been s t u d i e d . We i n t e n d t o c l a s s i f y a l l b l o c k i n g sets i n t h e l a r g e Mathieu systems S ( 5 , 8 , 2 4 ) , S ( 4 , 7 , 2 3 ) , S ( 3 , 6 , 2 2 ) and is S ( 2 , 5 , 2 1 ) i . e . PG(2,4). Each o f t h e s e case i s a s p e c i a l case and PG(2.4) o n l y t h e f i r s t . The s t u d y of b l o c k i n g sets i n o t h e r b i g Mathieu s y s t e m s w i l l be t h e a i m o f o u r n e x t p a p e r s . I n what f o l l o w s w e u s e t h e f o l l o w i n g n o t a t i o n s . D e n o t e by F t h e Fano ( B a e r ) s u b p l a n e of PG(2,4). F i x t h r e e non-concurrent l i n e s L,M,N.Put p=LnM, u € M / L , V E M / L , and suppose u , v , w E N . P u t A : =(L-(u})U(M - ( v } ) U{W)
.
The l i n e N i s c a l l e d t h e s h o r t s i d e o f A . Denote by H a t r i a n g l e w i t h o u t v e r t i c e s ; a l l h e r m i t i a n arcs are i s o m o r p h i c t o H. I n t h i s paper w e prove t h e f o l l o w i n g
in
PG(2,4)
1 . 4 THEOREM. I n PG(2,4) t h e b l o c k i n g s e t s S, I S I = k , w i t h n e c e s s a r i l y 7 1 k 5 1 4 , are e x a c t l y t h e f o l l o w i n g : ( a ) i f k=7, t h e n S=F; ( b ) i f k=8, t h e n e i t h e r S = F u {x) w i t h x @ S , o r S= A which i s i r r e d u c i b l e ; ( c ) i f k=9, t h e n e i t h e r S=H ( i r r e d u c i b l e ) , o r S = A U{x} w i t h X G A~ ; ( d ) if k=10, t h e n S = A U { x , y } , where x , y are n o t v e r t i c e s and t h e f o l l o w i n g c o n d i t i o n ho1ds:if x,y,w are c o l l i n e a r t h e n e i t h e r x , y , w , p o r x,y,w,u,v are collinear ( e ) Moreover, t h e b l o c k i n g sets w i t h I S l r l l are e x a c t l y t h e complements of t h o s e d e f i n e d above.
.
2. PROOF OF THE THEOREM By R e s u l t s 1.1, 1.2, 1.3 i t f o l l o w s t h a t i f S d e n o t e s a b l o c k i n g s e t i n PG(2,4), t h e n f o r any k w i t h 7 1 k S 1 4 t h e r e i s a t l e a s t one b l o c k i n g s e t . We n o t e t h a t i t i s enough t o c l a s s i f y t h e b l o c k i n g sets S w i t h 7 1 1 S 1 1 1 0 , s i n c e t h e o t h e r s are t h e complementary p o i n t - s e t s . Moreover, w e n o t e t h a t by 1.1 i t f o l l o w s 1.4 ( a ) and by 1.2 t h e f i r s t p a r t o f 1.4 ( c ) h a s been proved. I n t h e f o l l o w i n g Remark, a p a r t i c u l a r c o n s t r u c t i o n of a Fano s u b p l a n e F of PG(2,4) i s g i v e n . 2 . 1 REMARK. L e t C be a - ( q + l ) - a r c i n PG(2.q), t c C a p o i n t and T a l i n e l-sec a n t of C a t T. The s e t B:=CuT - { t } i s an i r r e d u c i b l e b l o c k i n g set i f q i s odd, cf.1171. I f q i s even and t h e knot of C i s denoted by k , t h e s e t B-{k} i s a b l o c k i n g set. I n t h e case of PG(2,4) s u c h a b l o c k i n g set i s t h e Fano p l a n e s i n c e i t c o n t a i n s 7 p o i n t s . Moreover, B and BUT-{k} are c l e a r l y i s o m o r p h i c , s i n c e C u { k } - { t } i s a c o n i c (cf.[ll], 8.4.1). I n t h e f o l l o w i n g w e u s e t h e n o t i o n of c h a r a c t e r s of a k - s e t K. W e denote by x , t h e number of l i n e s of PG(2,4) which are i - s e c a n t of s e t K . I t i s w e l l known [17] t h a t t h e f o l l o w i n g r e l a t i o n s h o l d xo+ (2.1)
+ X, +
Xi
x2 +
x3+
2x2 + 3x3 +
X4+
x 5 = 21
4 x 4 + 5 x S = 5k
2 x 2 + 6 x 3 + 1 2 x 4 + 2Oxs= k(k-1). Now w e prove 1.4 ( b ) , namely: 2.2 THEOREM. I n n = P G ( 2 , 4 ) t h e r e a r e o n l y two t y p e s (under up isomorphism) b l o c k i n g sets w i t h 8 p o i n t s : (a) the s e t A , t h a t is i r r e d u c i b l e , ( b ) t h e s e t F u ( x } , where F i s a Fano s u b p l a n e and x $ F .
of
Blocking Sets in the Projective Plane of Order Four with ICI=8.
PROOF - L e t C b e a b l o c k i n g set i n x,+x,+x,+x,
=
21, x , + ~ x , + ~ x ,= 19, x,+3x4 = 9
45
By (2.1) w e h a v e :
, so x q < 3
.
S t e p 1. W e p r o v e t h a t x,=2 or x , = l . Suppose x,=O. I t f o l l o w s t h a t x 3 = 9 , x,=1, x , = l l . D e n o t e by u , t h e number o f l i n e s t h r o u g h a f i x e d p o i n t o f C t h a t a r e i - s e c a n t of C. We h a v e : u,+u,+u, =5, u,+2u3=7. So, u p 0 f o r a n y p o i n t o f C , t h e n x z > l , a c o n t r a d i c t i o n . Suppose x,=3. Two 4 - s e c a n t l i n e s L,M are n e c e s s a r i l y i n c i d e n t a t a p o i n t o f C , o t h e r w i s e C i s n o t a b l o c k i n g set. Moreover, t h e t h i r d 4 - s e c a n t i n t e r sects L and M, and s o I C l 2 9 , a c o n t r a d i c t i o n . e p r o v e t h a t x , = l i m p l i e s C=F u { x ) . S t e p 2. W If x , = l , t h e n x,=6 , x,=4 , x,=lO. Denote by T t h e 4 - s e c a n t o f C and by t t h e p o i n t o f T o u t s i d e C. P u t C , : = C / T . We n o t e t h a t C ' : = C, u { t } i s a 5-arc o f JG. (Each l i n e # T t h r o u g h t i n t e r s e c t s C, and t h e n C , , a t e x a c t l y o n e p o i n t , and s o s u c h l i n e s a r e 2 - s e c a n t o f C ' ) . F i x P E T w i t h p f t . D e n o t e by u I t h e number o f l i n e s t h r o u g h p and i-sec a n t o f C. W e have u, +u,+u,+u,= 5, u2+2u,+3u,=7. W e h a v e u 4 = 1 and u , = 2 , ( i n f a c t x , = l i m p l i e s u,=l. I f u,=O o r 1 t h e n V p e T - { t ] w e h a v e a t m o s t 4 l i n e s t h a t a r e 3 - s e c a n t o f C , s i n c e e a c h 3 - s e c a n t i n t e r s e c t s T-{t}, w h i l e x , = l i m p l i e s x,=6). Then u,=O and u , = 2 . C o n s e q u e n t l y , a n y l i n e z T t h r o u g h a p o i n t p i n t e r s e c t s C a t 1 o r 3 p o i n t s and C, a t 0 o r 2 p o i n t s . T h e set C ' = C, u { t } i s o f t y p e ( 0 , 1 , 2 ) i . e . C ' i s a 5-arc of a . By [ l l ] (cf.8.4.1) C ' i s a c o n i c and C=FU{ x}.
S t e p 3. F i n a l l y , w e p r o v e t h a t x,=2 i m p l i e s C = A . The two 4 - s e c a n t s L,M a r e n e c e s s a r i l y i n c i d e n t a t a p o i n t of C. P u t u = L \ C and V E M \ C . S i n c e t h e l i n e c o n t a i n i n g u and v m u s t c o n t a i n a p o i n t w 6 C, i t follows C=A. 0 W e d e a l now w i t h t h e b l o c k i n g s e t s o f c a r d i n a l i t y 9 i n J G . W e use t h e n o t a t i o n s o f 1 . 4 and 2.1. I n 7t a h e r m i t i a n a r c H i s a n i r r e d u c i b l e b l o c k i n g 9-set a n d t h e 9 - s e t A u { x ) , where x i s n o t a v e r t e x o f A , i s c l e a r l y a r e d u c i b l e b l o c k i n g set. Note t h a t i n z t h e set ( F U { y ) ) U { x } i s a s p e c i a l case of t h e 9-set A U { x } , o b t a i n e d i f x i s c o l l i n e a r w i t h p a n d w. We p r o v e t h a t 2.3 THEOREM. I n n = P G ( 2 , 4 ) t h e r e a r e ( u n d e r up isomorphism) o n l y two sets o f 9 p o i n t s : t h e h e r m i t i a n arc and AU { x } w i t h x @ A
blocking
.
PROOF. L e t C be a b l o c k i n g s e t i n o f C. By ( 2 . 1 ) , w e h a v e :
x,+x,+x,+x,
=21
,
n with
x , + ~ x , + ~ x =, 24
,
I C I =9. Denote by xi t h e
x , + ~ x , = 12
. SO, x 5 4.
S t e p 1. Suppose x,=O, t h e n C i s a s e t o f t y p e (1,3) and b y r l l ] , s e t C i s a h e r m i t i a n a r c , s i n c e ICI=9. S t e p 2. Suppose x q = l , t h e n x,=9, x,=3, x , = 8 . Denote by w € C l i n e L which i s 4 - s e c a n t of C. We have: u,+u,+u,+l
=
5
t
characters
a
cf.
point
13.4.6, of
the
U2+2U,+3 = 8,
where u I i s t h e number of i - s e c a n t l i n e s t h r o u g h w € L . T h e s e i m p l y u , = us-1
,
u 2 = 5 - 2u,
.
It f o l l o w s t h a t u , = l o r 2. I n b o t h t h e s e cases u , 2 1 , and s o t h e r e i s a t l e a s t
L. Berardi and F. Eugeni
46 one 2-secant
through each of t h e four p o i n t s wcCnL.So x , 2 4 , a contradiction.
S t e p 3. Now w e p r o v e t h a t x,=2 i m p l i e s C= A U ( x } , where e i t h e r x i s a p o i n t on t h e s h o r t s i d e o f A or x is c o l l i n e a r w i t h p and W. Suppose x 4 = 2 , t h e n x 3 = 6 = x p r x l = 7 . S i n c e x , = 7 < l C I , t h e r e i s a p o i n t x which d o e s n o t l i e on a t a n g e n t l i n e . So C-{x) i s a b l o c k i n g set o f 8 p o i n t s , t h e n C -{x} = A o r F U { y } , y e F. Then x l i e s on t h e s h o r t s i d e o f A w i t h x @ { u , v , w } Suppose C -{,}=A. ( o t h e r w i s e t h e l i n e xw i s a t h i r d 4 - s e c a n t ) . I f w e c o m p l e t e A w i t h x i n s u c h a way, w e h a v e a b l o c k i n g s e t w i t h x,=2. Suppose C -{x) =F U { y } . Then x @ V u { y } , s i n c e C c o n t a i n s n o l i n e . I f w e c o m p l e t e F U {y) w i t h t h e r i x e d p o i n t x , :u.e h a v e a b l o c k i n g s e t w i t h xg =2. S i n c e F U ( x , y } i s t h e same a s A u { x ) when x,w and p are c o l l i n e a r , t h e a s s e r t i o n i s proved. S t e p 4 . Now w e p r o v e t h a t x,=3 i m p l i e s t h a t C = A U { x } , where x e u v and x , p , w are n o t c o l l i n e a r . Suppose x4 =3. Then x 3 = 3 , xp =9 , x1 =6. S i n c e x, = 6 < I C I, t h e r e i s a p o i n t x which d o e s n o t l i e o n a t a n g e n t l i n e . S o , C -{x} i s a b l o c k i n g s e t o f 8 p o i n t s . Suppose C - { x ) = F U { y ) Then ( c f . S t e p 3 a b o v e ) set C h a s xg =2 necessarily, a coiitradiction. Suppose C - { x ) = A . Then x e u v and x,p,w a r e n o t c o l l i n e a r ( o t h e r w i s e , by a b o v e S t e p 3, x 4 = 2 , a c o n t r a d i c t i o n ) . It f o l l o w s t h a t t h e l i n e xu i n t e r s e c t s C=A U { x } a t two p o i n t s of C , s i n c e i t d o e s n o t c o n t a i n p , t h e n xw i s t h e t h i r d 4-secant.
.
S t e p 5. F i n a l l y , we p r o v e t h a t x 4 = 4 i s i m p o s s i b l e . Two 4 - s e c a n t l i n e s L,M h a v e n e c e s s a r i l y o n e common p o i n t i n C. So, a t h i r d 4 - s e c a n t i n t e r s e c t s L and M a t two p o i n t s o f C. I f t h r e e 4-secants are c o n c u r r e n t , t h e n l C I 2 1 0 , a c o n t r a d i c t i o n . Suppose t h a t t h e t h i r d 4 - s e c a n t N of C i n t e r s e c t s L and M a t two d i s t i n c t p o i n t s of C . The f o u r t h 4 - s e c a n t h a s a t most t h r e e p o i n t s o f C i n common w i t h L,M,N. C o n s e q u e n t l y I C 1 2 1 0 , a c o n t r a d i c t i o n . S o , 2.3 i s c o m p l e t e l y proved. Now w e d e a l w i t h b l o c k i n g s e t s i n n w i t h 10 p o i n t s . I f x , y are p o i n t s o f rc ( w i t h t h e e x c e p t i o n of j u s t a few p o s i t i o n s f o r x and y ) t h e s e t A U { x , y } i s a b l o c k i n g set o f 10 p o i n t s . F o r e x a m p l e , i f z i s a v e r t e x of a h e r m i t i a n a r c H , t h e n t h e b l o c k i n g 10-set H U ( z } i s a s p e c i a l case of s e t A!J ( x , y } , o b t a i n e d by t a k i n g x , y on t h e s h o r t s i d e o f A Moreover, i f z d o e s n o t b e l o n g t o a s i d e of H, s e t H u { z ) i s i s o m o r p h i c t o a n a p p r o p r i a t e A { x , y ) w i t h x , y on t h e s h o r t s i d e . T h i s i s e a s y t o p r o v e . We c o n c l u d e t h i s s e c t i o n w i t h t h e f o l l o w i n g
.
u
2.4 THEOREM. I n n = P G ( 2 , 4 ) t h e b l o c k i n g sets of 10 p o i n t s ( n e c e s s a r i l y r e d u c i b l e ) a r e e x a c t l y t h e s e t s A U { x , y } , where x , y are n o t v e r t i c e s o f A and t h e f o l l o w i n g c o n d i t i o n h o l d s : i f x,y,w a r e c o l l i n e a r , t h e n t h e l i n e xy e i t h e r c o i n c i d e s w i t h t h e s h o r t s i d e o f A o r it c o n t a i n s p. PROOF. L e t C be a b l o c k i n g set i n n w i t h I C I = 10.Denote by x i t h e c h a r a c t e r s of C. By ( 2 . 1 ) w e o b t a i n x,=8-x,, x , = 3 ( x g - 1 ) , x,=16-3xg. S o , l I x , S 5 . I t i s o b v i o u s t h a t i f t w o 4 - s e c a n t L,M e x i s t , t h e y i n t e r s e c t a t a p o i n t o f C ( o t h e r w i s e ' I C l 2 1 1 ) . P u t p:=LnM, u : = L \ C , v : = M \ C ; d e n o t e by w a p o i n t o f C on t h e l i n e uv. Then t h e t r i a n g l e A = L U M U{w} - { u , v } i s c o n t a i n e d i n C . Put { x,y):=C \ A
.
S t e p 1. The case x q = 1 i s n o t p o s s i b l e . I f x,=1, t h e n x 3 = 1 3 , x,=O , x , = 7 . S o , t h e r e e x i s t s a p o i n t x E C t h r o u g h which no t a n g e n t p a s s e s . C o n s e q u e n t l y , i f u i i s t h e number o f i - s e c a n t - o f C t h r o u g h x , w e h a v e u , = u 2 = 0 , and s o u,+u,=5 and 2u3+3u,=9, which i m p l y u,=-l, a
Blocking Sets in the Projective Plane of Order Four
41
contradiction. S t e p 2. The case x,=2 i m p l i e s t h a t C= A U ( x , y } , where x , y l i e o n t h e short: s i d e uv o f A Suppose x4=2. Denote by u i t h e number o f l i n e s t h r o u g h p=LnM and i - s e c a n t of C. W e h a v e u1 +u, +u3+2 = 5 and u, +2u3+C = 9, from which u1 =u3 and u 2 = 3-2u3, c o n s e q u e n t l y u 3 = 1 o r 0. Suppose u 3 = l , t h e n u , = u , = l . I f x i s a p o i n t o n pw, t h e l i n e py i s t h e 2 - s e c a n t . I n t h i s case y i s on t h e s h o r t s i d e ; o t h e r w i s e yw i s a t h i r d & s e c a n t . But i n t h i s case xy i s a l s o a & s e c a n t , and t h i s is a c o n t r a d i c t i o n . T h i s l e a v e s u 3 = 0 , which i m p l i e s u,=O, u,=3. I n t h i s case pw, p x , py a r e t h r e e d i s t i n c t l i n e s . Moreover, x and y n e c e s s a r i l y l i e o n t h e s h o r t s i d e , o t h e r w i s e w e h a v e xq > 2 . So, t h e a s s e r t i o n of S t e p 2 i s proved.
.
S t e p 3. The case x,=3 i m p l i e s t h a t G A U { x , y } , where x and y are p o i n t s s a t i s f y one of t h e following conditions:
that
(i) X,YEPV (ii) x ~ u v , (iii) x ~ w p ,
y ~ p x y ~ x v (or x u ) . If x4=3, t h e n x 3 = 7 , x 2 = 6 , x , = 5 . Denote by N t h e t h i r d s e c a n t . It r e s u l t s v = N n u v . We h a v e two p o s s i b i l i t i e s : L,M and N a r e c o n c u r r e n t i n p o r n o t . F i r s t , s u p p o s e p = LflMflN.The p o i n t s o f C are a l l on L U M U N , so w , x , y ~ ? i s i n c e t h e y a r e n o t i n LUM. So C = A U { x , y ) w i t h c o n d i t i o n ( i ) . Next, s u p p o s e p = L n M g N . The p o i n t s o f C are i n LUMUN. P u t y : = C n ( N \ ( L U M ) ) - {w} and x :=C - ( A u ( y ) ) . Denote by u i t h e number of l i n e s which are i - s e c a n t o i C and c o n t a i n p.As i n t h e case of S t e p 2 , w e have: (I)
u3=ul=0
,
u,=3
(11) u3=u2 = u , = 1 .
or
The case ( I ) i m p l i e s a c o n t r a d i c t i o n . ( S i n c e i n AU{y} t h e l i n e s pw and py a r e 2 - s e c a n t o f C, t h e p o i n t x l i e s on t h e f i f t h l i n e t h r o u g h p. I n a n y case xw or xy i s a n u l t e r i o r 4 - s e c a n t ) . F i n a l l y , s u p p o s e t h a t (11) h o l d s . S i n c e i n A U { y } , t h e l i n e s pw and py are 2 - s e c a n t o f C, t h e p o i n t x l i e s on o n e of them. I f X E py, n e c e s s a r i l y x E uv; o t h e r w i s e xw i s a 4 - s e c a n t . T h i s i s t h e case (ii). I f x E pw, t h e l i n e xy n e c e s s a r i l y c o n t a i n s e i t h e r u or v , o t h e r w i s e xy s h o u l d b e a & s e c a n t . T h i s i s t h e case (iii),So, t h e proof of S t e p 3 is f i n i s h e d . S t e p 4. The case x,=4 i m p l i e s t h a t C = A U { x , y } , where x , y s a t i s f y one of t h e following c o n d i t i o n s :
(1)
PEXY
(2)
UEXU
(3) (4)
, WBXY
,
X,Y
(or v ~ x y )
,
XEUV
, w,pexy
;
xEwp
,
points
which
B uv ;
x,y$A
u,v$Xy#pw
are
,
;
yeuv.
Suppose x 4 = 0 and d e n o t e by u i t h e number of i - s e c a n t c o n t a i n i n g p. As i n S t e p s 2 and 3, w e h a v e e i t h e r ( a j u l = u 3 = O = u , or (b) u,=u,=u,=l. I n t h e case ( a ) t h e l i n e s pw, px, py,L,M are a l l d i s t i n c t . I t i s t r i v i a l t h a t e i t h e r xy c o n t a i n s u o r x ~ u v ;o t h e r w i s e x 4 > 4 . So, w e a r e i n s i t u a t i o n s ( 2 ) and ( 3 ) f o r x and y. w i t h y on I n t h e case ( b ) w e n e c e s s a r i l y h a v e e i t h e r p c x y # pw o r x ~ p w o n e o f two r e m a i n i n g l i n e s t h r o u g h p. Moreover, yx c o n t a i n s n e i t h e r u n o r v and y e u v . T h i s c o r r e s p o n d s t o s i t u a t i o n s (1) and ( 4 ) f o r x and y. So, S t e p 4 i s proved. S t e p 5. The case x 4 = 5 i s i m p o s s i b l e .
L. Berardi and F. Eugeni
48
I f x,=5,we h a v e x , = l , x z = 1 2 , x , = 3 . L e t R b e t h e o n l y 3 - s e c a n t , t a k e h E RnC. I f u , a r e t h e number o f i - s e c a n t c o n t a i n i n g h,we have: u,+u,+l+u,=5,uz+2+3u, =9 and r h e n u, =2u, -3 , uz =7-3u,, from which u,=2, uz =u, =1. C o n s e q u e n t l y , e a c h o f t h r e e p o i n t s o n C n R i s i n c i d e n t w i t h e x a c t l y two 4 - s e c a n t . Then x, 2 6 , a c o n t r a d i c t i o n . S t e p 6. As a l a s t r e m a r k , w e n o t e t h a t f o r A U { x , y } e v e r y case i s p r e s e n t e d i n t h e a b o v e s t e p s , e x c e p t t h e f o l l o w i n g : xy # u v , w ~ x y ,p @xy. Note t h a t i n t h i s e x c e p t i o n s xy i s e n t i r e l y c o n t a i n e d i n A U { x , y } which, i n t h i s case, i s n o t a b l o c k i n g s e t . S i n c e t h i s case i s e x c l u d e d by t h e h y p o t h e s i s o f t h e t h e o r e m , t h e a s s e r t i o n 2.4 i s c o m p l e t e l y proved.
3. FINAL REMARKS I n t h i s S e c t i o n w e d e a l about o t h e r q u e s t i o n s connected with blocking sets. We r e c a l l t h a t a n n - f o l d b l o c k i n g s e t i n a n i n c i d e n c e s t r u c t u r e i s t h e u n i o n o f n d i s j o i n t b l o c k i n g sets. I t i s w e l l known ( c f . [ll] ) t h a t i n PG(2,q), q a s q u a r e , t h e r e e x i s t s a Baer p a r t i t i o n i n q-Sq+l Baer s u b p l a n e s . Such a p a r t i t i o n , w h e n q = 4 , i s a 3 - f o l d b l o c k i n g s e t t h a t w e c a l l a Fano p a r t i t i o n . 3.1 REMARK. Denote by 58 a n n - f o l d b l o c k i n g s e t i n PG(2,4), t h e n 1113. Namely: ( a ) n=2, and 9 i s t h e u n i o n e i t h e r o f one b l o c k i n g s e t a n d i t s complement o r o f two b l o c k i n g sets c o n t a i n e d i n e a c h o f them. ( b ) n=3, and 9 i s a Fano p a r t i t i o n .
Another matter i s t h e o p e n problem o f t h e e x i s t e n c e o f b l o c k i n g s e t s i n p r o j e c t i v e o r a f f i n e s p a c e s o f small o r d e r . R e c e n t l y Rajola [ 2 5 ] h a s g i v e n a n example o f b l o c k i n g s e t i n PG(3,5). S o , when q 1 5 , t h e problem i s open i n PG(r.5) ( 1 - 2 4 ) . AG(r.5) ( r 2 3 , c f . [16]) and i n P G ( r , 4 ) (1-23). I n AG(r.4) ( 1 - 2 3 ) no b l o c k i n g set t h e r e e x i s t s . We hope t h a t t h e c l a s s i f i c a t i o n o f b l o c k i n g s e t s i n PG(2.4) c o u l d be u s e f u l f o r t h i s problem. Moreover w e aim t o g i v e a c o m p l e t e c l a s s i f i c a t i o n o f b l o c k i n g sets i n p r o j e c t i v e or a f f i n e s p a c e s o f o r d e r 9 1 9 . S e e t h e l a s t p a r t o f [19] i n t h e s e P r o c e e d i n g s . The l a s t problem t h a t w e want t o r e m a r k i n t h i s S e c t i o n i s t h e problem o f t h e i n d e x ( c f . [ 221, [24] ).Denote by B a b l o c k i n g s e t i n a n i n c i d e n c e s t r u c t u r e . The i n d e x i = i ( B ) of B i s t h e smallest number of b l o c k s s u c h t h a t t h e u n i o n of them c o n t a i n s B. I n [22], i t h a s b e e n proved t h a t i n a p r o j e c t i v e of o r d e r q p l a n e w e h a v e 3 l i ( B ) l q + l f o r a n y b l o c k i n g s e t . I n P G ( 2 , 4 ) , w i t h r e g a r d t o 1 . 4 Theorem o f S e c t i o n 1, w e h a v e t h e f o l l o w i n g s i t u a t i o n . ( a ) A Fano s u b p l a n e F, s e t F u { x } w i t h x on a 3 - s e c a n t , a t r i a n g l e A , a h e r m i t i a n a r c H and t h e b l o c k i n g s e t s o f t y p e e i t h e r A u { x } o r A u { x , y } w i t h x,y w c o l l i n e a r , have a l l index t h r e e . ( b ) E i t h e r F G ( x ) w i t h x o u t s i d e o n e 3 - s e c a n t or A u { x , y ) w i t h x,y,w non- c o l l i n e a r have index four.
It r e m a i n s t o c l a s s i f y t h e i n d e x of t h e complements o f a b o v e c o n s i d e r e d b l o c k i n g sets which are r e d u c i b l e . T h i s i s a t r i v i a l e x e r c i s e , w e remark o n l y t h a t o n e o f them, namely t h e complement of t h e Fano s u b p l a n e h a s i n d e x 5 n e c e s s a r i l y , soeach i n d e x i s p o s s i b l e . REFERENCES
[ 13
L.Berardi-F.Eugeni, On t h e c a r d i n a l i t y o f b l o c k i n g s e t s i n PG(2,q). J. o f Geometry 22 ( 1 9 8 4 ) , 5-14.
[ 2 ] L.Berardi-F.Eugeni-O.Ferri, Boll.Un.Mat.Ital.,
sez.
S u i b l o c k i n g sets D, 1 ( 1 9 8 4 ) , 141-164.
nei
sistemi
di
Steiner.
Blocking Sets in the Projective Plane of Order Four
c 31 c 41
49
A.Beutelspacher-F.Eugeni, On blocking sets in projective and affine spaces of large order. Communicated to "Grundlagen der Geometrie" Oberwolfach, 27-10/2-11-1985. Rend.di Mat. Roma, to appear.
A.Bruen, Blocking sets in finite projective planes. SIAM J.App1. 21 (1971), 380-392.
Math.,
Baer subplanes and blocking sets. Bull.Amer.Math.Soc.,76 (1970), r 51 A.Bruen, 342-344. c 61 A.Bruen-A.Thas, Blocking sets. Geom. Dedicata, 6 (1977). 193-203.
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P.Cameron-F.Mazzocca, Bijections which Dedicata, to appear.
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J.Di Paola, On a restricted class of block design games., Canad. J.Math., 18 (1966), 225-236.
c93 J.Di Paola, On minimum blocking coalitions in small projective plane games. SIAM J. Appl. Math., 17 (1969), 378-392.
[lo] M.
de Resmini, On k-sets of class [O,q/2-1, q/2 q/Z+l,q] in a even order q. Europ.J.Combin.. 6 (1985). 303-315.
plane of
[ 111 J.W.P.Hirschfeld, Projective geometries over finite fields. Clarendon Press, Oxford, 1979.
[12] A.J.Hoffman-M.Richardson, 1 10-1 28.
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c131 W.Jdnsson, On the Mathieu groups M,, ,M,, , M,, and the uniqueness of associated Steiner systems. Math. Z. 125 (1972), 193-214. [ I 4 1 F.Mazzocca-G.Tallini, On the non existence of blocking-sets in PG(n,q) and AG(n,q), for all large enough n. Simon Stevin, 1 (1985), 43-50.
1151 M.Richardson, On
finite projective games. Proc.
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7,
(1956). 458-465.
[I63 G.Tallini, Problemi e risultati nelle geometrie di Galois. Relaz.
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c17-J G.Tallini, k-insiemi e blocking sets in PG(r,q) e in AG(r,q).
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(1973) 1st. Mat. Univ. Napoli.
Comb. Fac. Ing. Univ. L'Aquila, 1 (1982).
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[18] G.Tallini, Blocking sets nei sistemi di
PG(r,q) ed AG(r,q).
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p 9 - j G.Tallini, On blocking sets in finite projective and
affine spaces.
[20] J.A.Tood, A representation of the Mathieu group M, group. Ann. Mat.Pura e Appl.. 71 (1966), 199-238.
as a
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In
collineation
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3d.
Supplementary references on Section 3.
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L. Berardi and F. Eugeni re riguardo a l l ' i n d i c e tre. Boll.Un.Mat.Ital.,
4-A(1985),
441-450.
[23]
A.Beutelspacher-F.Eugeni, On n - f o l d b l o c k i n g sets. A n n a l s of Discrete M a t h e m a t i c s 30 ( 1 9 8 6 ) , 31-38. Proceedings of I n t e r n a t i o n a l Conference "Combinat or i c s '84"-Bari.
[24]
F . E u g e n i , E. Mayer, On b l o c k i n g sets o f i n d e x two, I n t h e s e P r o c e e d i n g s .
[25]
A . R a j o l a , Un e s e m p i o d i b l o c k i n g s e t i n PG(3.q) p e r o g n i q 2 5 . Quad.n.57 (1985), Sem. Geom. Comb. Univ. Roma "La S a p i e n z a " . I n t h e s e P r o c e e d i n g s .
1 2 6 3 C.O'Keefe-A.Venezia, B l o c k i n g sets i n AG(r.5). Comb. Univ. Roma "La S a p i e n z a " .
L. B e r a r d i Dipartimento d i Ingegneria E l e t t r i c a S e m i n a r i o d i Geometria C o m b i n a t o r i a L'Aquila ( I t a l y )
Quad. n .
F. Eugeni
56,
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I s t i t u t o d i Scienze Facolta' d i Architettura
Chieti (Italy)
Geom.
43
BLOCKING SETS I N THE PROJECTIVE PLANE OF ORDER FOUR L u i g i a BERARDI and F r a n c o EUGENI Dipartimento d i Ingegneria E l e t t r i c a . U n i v e r s i t a ' d e L'Aquila ( I t a l i a ) I s t i t u t o d i Scienze. Universita' d i C h i e t i (Italia) We g i v e a c o m p l e t e c h a r a c t e r i z a t i o n o f b l o c k i n g s e t s i n PG(2,4).
1. INTRODUCTION A blocking set i n a f i n i t e i n c i d e n c e s t r u c t u r e ( P * 9 ,I) is a s u b s e t S o f P s u c h t h a t e v e r y b l o c k o f $ i s i n c i d e n t t o S and t o P-S a t l e a s t i n o n e p o i n t . The d e f i n i t i o n i m p l i e s t h a t i f S i s a b l o c k i n g s e t , t h e n P-S i s a b l o c k i n g s e t t o o . A b l o c k i n g s e t i s s a i d t o b e i r r e d u c i b l e i f v x E S t h e s e t S- ( x } i s n o t a blocking set. The c o n c e p t o f b l o c k i n s e t was i n t r o d u c e d , f o l l o w i n g a n i d e a o f Von Newmann and M o r g e n s t e r n (L205 , f o o t o n o t e 3, p. 4 6 9 ) , w i t h t h e name b l o c k i n g coal i t i o n s by M.Richardson i n 1956 [16], i n r e l a t i o n t o t h e s o - c a l l e d finite p r o j e c t i v e games (see a l s o [12]). I n 1966 J . D i P a o l a wrote two p a p e r s on t h e s e q u e s t i o n s w i t h t h e f i r s t r e s u l t s - i g p r o j e c t i v e p l a n e s of small o r d e r and i n t h e d u a l o f a S t e i n e r s y s t e m , [8J, L9J. I n 1970 A.Bruen i n t r o d u c e d t h e word " b l o c k i n g set" and began a sistematic s t u d y . S i n c e t h e n s e v e r a l a u t h o r s h a v e c o n t r i b u t e d t o t h i s s t u d y . R e c e n t l y i t h a s b e e n proved t h a t t h e e x i s t e n c e of b l o c k i n g sets i n p r o j e c t i v e o r a f f i n e s p a c e s o f h i g h d i m e n s i o n whoFe o r d e r i s l a r g e enough ( c f . [3], 171 , [14]).A u s e f u l l i s t c a n b e found i n [ 1 9 J . ~ erecall:
1.1 RESULT ( B r u e n [4], [S]). L e t S b e a b l o c k i n g set i n a p r o j e c t i v e p l a n e of o r d e r q.Then: q+Sq+l I S I q2-&j. Moreover, e q u a l i t y on t h e l e f t h a n d - s i d e h o l d s i f a n d o n l y i f S i s a B a e r s u b p l a n e . (On t h e r i g h t h a n d - s i d e i f f S is t h e complement of a B a e r s u b p l a n e ) . 1 . 2 RESULT (Bruen-Thas [6] , T a l l i n i [17] , [19] ). L e t S a n i r r e d u c i b l e s e t i n a p r o j e c t i v e p l a l i e o f o r d e r q . Then S I q S q + l , moreover t h e h o l d s i f and o n l y i f S i s a h e r m i t i a n arc.
blocking equality
S u p p o s e q = p h , p a p r i m e , q > 2 . We d e f i n e t h e f u n c t i o n m(q) as f o l l o w s : m ( q ) = G , i f q i s a s q u a r e ; m ( q ) = ( q + l ) / 2 , i f q i s a p r i m e and m(q)=ph-d, i f h > l is odd and where d d e n o t e s t h e g r e a t e s t d i v i s o r of h , d i f f e r e n t f r o m h. Then
1.3 RESULT ( B e r a r d i - E u g e n i [ I]). F o r any i n t e g e r k w i t h q+m(q)+l I k 5 q2 - m(q), t h e r e e x i s t s a b l o c k i n g s e t i n PG(2,q) h a v i n g e x a c t l y k e l e m e n t s . Many i n f o r m a t i o n s a b o u t sets o f PG(2,q). q e v e n , . i n t e r s e c t e d by a l i n e 0, q / 2 - 1, q / 2 , q/2+1 o r q p o i n t s , c a n b e found i n L l O ] .
in
The a i m o f t h i s p a p e r i s t o s t u d y t h e b l o c k i n g s e t s i n PG(2,4). Why j u s t t h i s c a s e ? We h a v e s e v e r a l r e a s o n s . a ) I f q=2, t h e r e i s no b l o c k i n g set i n PG(2,2); i f q = 3 w e h a v e i n PG(2.3) So q=4 i s r e a l l y t h e o n l y two b l o c k i n g s e t s up t o isomorphism (see [ 173). f i r s t case; moreover f o r a n y k w i t h 7 I k 1 1 4 w e h a v e a t least o n e b l o c k i n g set w i t h k p o i n t s , by 1.3. b) The s p a c e PG(3,4) i s t h e o n l y 3 - d i m e n s i o n a l case i n which t h e e x i s t e n c e of b l o c k i n g s e t s i s a n open problem. W e h o p e t h a t t h e s t u d y o f b l o c k i n g sets of PG(3,4) c a n depend on t h e i r s e c t i o n s w i t h a p l a n e t h a t are t h e b l o c k i n g sets o f FG(2,4). C ) I n [ 2 ] and [24] a l l b l o c k i n g sets i n t h e l i t t l e M a t h i e u s y s t e m s
L. Berwdi and F. Eugeni
44
S ( 5 , 6 , 1 2 ) and i t s c o n t r a c t i o n s have been s t u d i e d . We i n t e n d t o c l a s s i f y a l l b l o c k i n g sets i n t h e l a r g e Mathieu systems S ( 5 , 8 , 2 4 ) , S ( 4 , 7 , 2 3 ) , S ( 3 , 6 , 2 2 ) and is S ( 2 , 5 , 2 1 ) i . e . PG(2,4). Each o f t h e s e case i s a s p e c i a l case and PG(2.4) o n l y t h e f i r s t . The s t u d y of b l o c k i n g sets i n o t h e r b i g Mathieu s y s t e m s w i l l be t h e a i m o f o u r n e x t p a p e r s . I n what f o l l o w s w e u s e t h e f o l l o w i n g n o t a t i o n s . D e n o t e by F t h e Fano ( B a e r ) s u b p l a n e of PG(2,4). F i x t h r e e non-concurrent l i n e s L,M,N.Put p=LnM, u € M / L , V E M / L , and suppose u , v , w E N . P u t A : =(L-(u})U(M - ( v } ) U{W)
.
The l i n e N i s c a l l e d t h e s h o r t s i d e o f A . Denote by H a t r i a n g l e w i t h o u t v e r t i c e s ; a l l h e r m i t i a n arcs are i s o m o r p h i c t o H. I n t h i s paper w e prove t h e f o l l o w i n g
in
PG(2,4)
1 . 4 THEOREM. I n PG(2,4) t h e b l o c k i n g s e t s S, I S I = k , w i t h n e c e s s a r i l y 7 1 k 5 1 4 , are e x a c t l y t h e f o l l o w i n g : ( a ) i f k=7, t h e n S=F; ( b ) i f k=8, t h e n e i t h e r S = F u {x) w i t h x @ S , o r S= A which i s i r r e d u c i b l e ; ( c ) i f k=9, t h e n e i t h e r S=H ( i r r e d u c i b l e ) , o r S = A U{x} w i t h X G A~ ; ( d ) if k=10, t h e n S = A U { x , y } , where x , y are n o t v e r t i c e s and t h e f o l l o w i n g c o n d i t i o n ho1ds:if x,y,w are c o l l i n e a r t h e n e i t h e r x , y , w , p o r x,y,w,u,v are collinear ( e ) Moreover, t h e b l o c k i n g sets w i t h I S l r l l are e x a c t l y t h e complements of t h o s e d e f i n e d above.
.
2. PROOF OF THE THEOREM By R e s u l t s 1.1, 1.2, 1.3 i t f o l l o w s t h a t i f S d e n o t e s a b l o c k i n g s e t i n PG(2,4), t h e n f o r any k w i t h 7 1 k S 1 4 t h e r e i s a t l e a s t one b l o c k i n g s e t . We n o t e t h a t i t i s enough t o c l a s s i f y t h e b l o c k i n g sets S w i t h 7 1 1 S 1 1 1 0 , s i n c e t h e o t h e r s are t h e complementary p o i n t - s e t s . Moreover, w e n o t e t h a t by 1.1 i t f o l l o w s 1.4 ( a ) and by 1.2 t h e f i r s t p a r t o f 1.4 ( c ) h a s been proved. I n t h e f o l l o w i n g Remark, a p a r t i c u l a r c o n s t r u c t i o n of a Fano s u b p l a n e F of PG(2,4) i s g i v e n . 2 . 1 REMARK. L e t C be a - ( q + l ) - a r c i n PG(2.q), t c C a p o i n t and T a l i n e l-sec a n t of C a t T. The s e t B:=CuT - { t } i s an i r r e d u c i b l e b l o c k i n g set i f q i s odd, cf.1171. I f q i s even and t h e knot of C i s denoted by k , t h e s e t B-{k} i s a b l o c k i n g set. I n t h e case of PG(2,4) s u c h a b l o c k i n g set i s t h e Fano p l a n e s i n c e i t c o n t a i n s 7 p o i n t s . Moreover, B and BUT-{k} are c l e a r l y i s o m o r p h i c , s i n c e C u { k } - { t } i s a c o n i c (cf.[ll], 8.4.1). I n t h e f o l l o w i n g w e u s e t h e n o t i o n of c h a r a c t e r s of a k - s e t K. W e denote by x , t h e number of l i n e s of PG(2,4) which are i - s e c a n t of s e t K . I t i s w e l l known [17] t h a t t h e f o l l o w i n g r e l a t i o n s h o l d xo+ (2.1)
+ X, +
Xi
x2 +
x3+
2x2 + 3x3 +
X4+
x 5 = 21
4 x 4 + 5 x S = 5k
2 x 2 + 6 x 3 + 1 2 x 4 + 2Oxs= k(k-1). Now w e prove 1.4 ( b ) , namely: 2.2 THEOREM. I n n = P G ( 2 , 4 ) t h e r e a r e o n l y two t y p e s (under up isomorphism) b l o c k i n g sets w i t h 8 p o i n t s : (a) the s e t A , t h a t is i r r e d u c i b l e , ( b ) t h e s e t F u ( x } , where F i s a Fano s u b p l a n e and x $ F .
of
Blocking Sets in the Projective Plane of Order Four with ICI=8.
PROOF - L e t C b e a b l o c k i n g set i n x,+x,+x,+x,
=
21, x , + ~ x , + ~ x ,= 19, x,+3x4 = 9
45
By (2.1) w e h a v e :
, so x q < 3
.
S t e p 1. W e p r o v e t h a t x,=2 or x , = l . Suppose x,=O. I t f o l l o w s t h a t x 3 = 9 , x,=1, x , = l l . D e n o t e by u , t h e number o f l i n e s t h r o u g h a f i x e d p o i n t o f C t h a t a r e i - s e c a n t of C. We h a v e : u,+u,+u, =5, u,+2u3=7. So, u p 0 f o r a n y p o i n t o f C , t h e n x z > l , a c o n t r a d i c t i o n . Suppose x,=3. Two 4 - s e c a n t l i n e s L,M are n e c e s s a r i l y i n c i d e n t a t a p o i n t o f C , o t h e r w i s e C i s n o t a b l o c k i n g set. Moreover, t h e t h i r d 4 - s e c a n t i n t e r sects L and M, and s o I C l 2 9 , a c o n t r a d i c t i o n . e p r o v e t h a t x , = l i m p l i e s C=F u { x ) . S t e p 2. W If x , = l , t h e n x,=6 , x,=4 , x,=lO. Denote by T t h e 4 - s e c a n t o f C and by t t h e p o i n t o f T o u t s i d e C. P u t C , : = C / T . We n o t e t h a t C ' : = C, u { t } i s a 5-arc o f JG. (Each l i n e # T t h r o u g h t i n t e r s e c t s C, and t h e n C , , a t e x a c t l y o n e p o i n t , and s o s u c h l i n e s a r e 2 - s e c a n t o f C ' ) . F i x P E T w i t h p f t . D e n o t e by u I t h e number o f l i n e s t h r o u g h p and i-sec a n t o f C. W e have u, +u,+u,+u,= 5, u2+2u,+3u,=7. W e h a v e u 4 = 1 and u , = 2 , ( i n f a c t x , = l i m p l i e s u,=l. I f u,=O o r 1 t h e n V p e T - { t ] w e h a v e a t m o s t 4 l i n e s t h a t a r e 3 - s e c a n t o f C , s i n c e e a c h 3 - s e c a n t i n t e r s e c t s T-{t}, w h i l e x , = l i m p l i e s x,=6). Then u,=O and u , = 2 . C o n s e q u e n t l y , a n y l i n e z T t h r o u g h a p o i n t p i n t e r s e c t s C a t 1 o r 3 p o i n t s and C, a t 0 o r 2 p o i n t s . T h e set C ' = C, u { t } i s o f t y p e ( 0 , 1 , 2 ) i . e . C ' i s a 5-arc of a . By [ l l ] (cf.8.4.1) C ' i s a c o n i c and C=FU{ x}.
S t e p 3. F i n a l l y , w e p r o v e t h a t x,=2 i m p l i e s C = A . The two 4 - s e c a n t s L,M a r e n e c e s s a r i l y i n c i d e n t a t a p o i n t of C. P u t u = L \ C and V E M \ C . S i n c e t h e l i n e c o n t a i n i n g u and v m u s t c o n t a i n a p o i n t w 6 C, i t follows C=A. 0 W e d e a l now w i t h t h e b l o c k i n g s e t s o f c a r d i n a l i t y 9 i n J G . W e use t h e n o t a t i o n s o f 1 . 4 and 2.1. I n 7t a h e r m i t i a n a r c H i s a n i r r e d u c i b l e b l o c k i n g 9-set a n d t h e 9 - s e t A u { x ) , where x i s n o t a v e r t e x o f A , i s c l e a r l y a r e d u c i b l e b l o c k i n g set. Note t h a t i n z t h e set ( F U { y ) ) U { x } i s a s p e c i a l case of t h e 9-set A U { x } , o b t a i n e d i f x i s c o l l i n e a r w i t h p a n d w. We p r o v e t h a t 2.3 THEOREM. I n n = P G ( 2 , 4 ) t h e r e a r e ( u n d e r up isomorphism) o n l y two sets o f 9 p o i n t s : t h e h e r m i t i a n arc and AU { x } w i t h x @ A
blocking
.
PROOF. L e t C be a b l o c k i n g s e t i n o f C. By ( 2 . 1 ) , w e h a v e :
x,+x,+x,+x,
=21
,
n with
x , + ~ x , + ~ x =, 24
,
I C I =9. Denote by xi t h e
x , + ~ x , = 12
. SO, x 5 4.
S t e p 1. Suppose x,=O, t h e n C i s a s e t o f t y p e (1,3) and b y r l l ] , s e t C i s a h e r m i t i a n a r c , s i n c e ICI=9. S t e p 2. Suppose x q = l , t h e n x,=9, x,=3, x , = 8 . Denote by w € C l i n e L which i s 4 - s e c a n t of C. We have: u,+u,+u,+l
=
5
t
characters
a
cf.
point
13.4.6, of
the
U2+2U,+3 = 8,
where u I i s t h e number of i - s e c a n t l i n e s t h r o u g h w € L . T h e s e i m p l y u , = us-1
,
u 2 = 5 - 2u,
.
It f o l l o w s t h a t u , = l o r 2. I n b o t h t h e s e cases u , 2 1 , and s o t h e r e i s a t l e a s t
L. Berardi and F. Eugeni
46 one 2-secant
through each of t h e four p o i n t s wcCnL.So x , 2 4 , a contradiction.
S t e p 3. Now w e p r o v e t h a t x,=2 i m p l i e s C= A U ( x } , where e i t h e r x i s a p o i n t on t h e s h o r t s i d e o f A or x is c o l l i n e a r w i t h p and W. Suppose x 4 = 2 , t h e n x 3 = 6 = x p r x l = 7 . S i n c e x , = 7 < l C I , t h e r e i s a p o i n t x which d o e s n o t l i e on a t a n g e n t l i n e . So C-{x) i s a b l o c k i n g set o f 8 p o i n t s , t h e n C -{x} = A o r F U { y } , y e F. Then x l i e s on t h e s h o r t s i d e o f A w i t h x @ { u , v , w } Suppose C -{,}=A. ( o t h e r w i s e t h e l i n e xw i s a t h i r d 4 - s e c a n t ) . I f w e c o m p l e t e A w i t h x i n s u c h a way, w e h a v e a b l o c k i n g s e t w i t h x,=2. Suppose C -{x) =F U { y } . Then x @ V u { y } , s i n c e C c o n t a i n s n o l i n e . I f w e c o m p l e t e F U {y) w i t h t h e r i x e d p o i n t x , :u.e h a v e a b l o c k i n g s e t w i t h xg =2. S i n c e F U ( x , y } i s t h e same a s A u { x ) when x,w and p are c o l l i n e a r , t h e a s s e r t i o n i s proved. S t e p 4 . Now w e p r o v e t h a t x,=3 i m p l i e s t h a t C = A U { x } , where x e u v and x , p , w are n o t c o l l i n e a r . Suppose x4 =3. Then x 3 = 3 , xp =9 , x1 =6. S i n c e x, = 6 < I C I, t h e r e i s a p o i n t x which d o e s n o t l i e o n a t a n g e n t l i n e . S o , C -{x} i s a b l o c k i n g s e t o f 8 p o i n t s . Suppose C - { x ) = F U { y ) Then ( c f . S t e p 3 a b o v e ) set C h a s xg =2 necessarily, a coiitradiction. Suppose C - { x ) = A . Then x e u v and x,p,w a r e n o t c o l l i n e a r ( o t h e r w i s e , by a b o v e S t e p 3, x 4 = 2 , a c o n t r a d i c t i o n ) . It f o l l o w s t h a t t h e l i n e xu i n t e r s e c t s C=A U { x } a t two p o i n t s of C , s i n c e i t d o e s n o t c o n t a i n p , t h e n xw i s t h e t h i r d 4-secant.
.
S t e p 5. F i n a l l y , we p r o v e t h a t x 4 = 4 i s i m p o s s i b l e . Two 4 - s e c a n t l i n e s L,M h a v e n e c e s s a r i l y o n e common p o i n t i n C. So, a t h i r d 4 - s e c a n t i n t e r s e c t s L and M a t two p o i n t s o f C. I f t h r e e 4-secants are c o n c u r r e n t , t h e n l C I 2 1 0 , a c o n t r a d i c t i o n . Suppose t h a t t h e t h i r d 4 - s e c a n t N of C i n t e r s e c t s L and M a t two d i s t i n c t p o i n t s of C . The f o u r t h 4 - s e c a n t h a s a t most t h r e e p o i n t s o f C i n common w i t h L,M,N. C o n s e q u e n t l y I C 1 2 1 0 , a c o n t r a d i c t i o n . S o , 2.3 i s c o m p l e t e l y proved. Now w e d e a l w i t h b l o c k i n g s e t s i n n w i t h 10 p o i n t s . I f x , y are p o i n t s o f rc ( w i t h t h e e x c e p t i o n of j u s t a few p o s i t i o n s f o r x and y ) t h e s e t A U { x , y } i s a b l o c k i n g set o f 10 p o i n t s . F o r e x a m p l e , i f z i s a v e r t e x of a h e r m i t i a n a r c H , t h e n t h e b l o c k i n g 10-set H U ( z } i s a s p e c i a l case of s e t A!J ( x , y } , o b t a i n e d by t a k i n g x , y on t h e s h o r t s i d e o f A Moreover, i f z d o e s n o t b e l o n g t o a s i d e of H, s e t H u { z ) i s i s o m o r p h i c t o a n a p p r o p r i a t e A { x , y ) w i t h x , y on t h e s h o r t s i d e . T h i s i s e a s y t o p r o v e . We c o n c l u d e t h i s s e c t i o n w i t h t h e f o l l o w i n g
.
u
2.4 THEOREM. I n n = P G ( 2 , 4 ) t h e b l o c k i n g sets of 10 p o i n t s ( n e c e s s a r i l y r e d u c i b l e ) a r e e x a c t l y t h e s e t s A U { x , y } , where x , y are n o t v e r t i c e s o f A and t h e f o l l o w i n g c o n d i t i o n h o l d s : i f x,y,w a r e c o l l i n e a r , t h e n t h e l i n e xy e i t h e r c o i n c i d e s w i t h t h e s h o r t s i d e o f A o r it c o n t a i n s p. PROOF. L e t C be a b l o c k i n g set i n n w i t h I C I = 10.Denote by x i t h e c h a r a c t e r s of C. By ( 2 . 1 ) w e o b t a i n x,=8-x,, x , = 3 ( x g - 1 ) , x,=16-3xg. S o , l I x , S 5 . I t i s o b v i o u s t h a t i f t w o 4 - s e c a n t L,M e x i s t , t h e y i n t e r s e c t a t a p o i n t o f C ( o t h e r w i s e ' I C l 2 1 1 ) . P u t p:=LnM, u : = L \ C , v : = M \ C ; d e n o t e by w a p o i n t o f C on t h e l i n e uv. Then t h e t r i a n g l e A = L U M U{w} - { u , v } i s c o n t a i n e d i n C . Put { x,y):=C \ A
.
S t e p 1. The case x q = 1 i s n o t p o s s i b l e . I f x,=1, t h e n x 3 = 1 3 , x,=O , x , = 7 . S o , t h e r e e x i s t s a p o i n t x E C t h r o u g h which no t a n g e n t p a s s e s . C o n s e q u e n t l y , i f u i i s t h e number o f i - s e c a n t - o f C t h r o u g h x , w e h a v e u , = u 2 = 0 , and s o u,+u,=5 and 2u3+3u,=9, which i m p l y u,=-l, a
Blocking Sets in the Projective Plane of Order Four
41
contradiction. S t e p 2. The case x,=2 i m p l i e s t h a t C= A U ( x , y } , where x , y l i e o n t h e short: s i d e uv o f A Suppose x4=2. Denote by u i t h e number o f l i n e s t h r o u g h p=LnM and i - s e c a n t of C. W e h a v e u1 +u, +u3+2 = 5 and u, +2u3+C = 9, from which u1 =u3 and u 2 = 3-2u3, c o n s e q u e n t l y u 3 = 1 o r 0. Suppose u 3 = l , t h e n u , = u , = l . I f x i s a p o i n t o n pw, t h e l i n e py i s t h e 2 - s e c a n t . I n t h i s case y i s on t h e s h o r t s i d e ; o t h e r w i s e yw i s a t h i r d & s e c a n t . But i n t h i s case xy i s a l s o a & s e c a n t , and t h i s is a c o n t r a d i c t i o n . T h i s l e a v e s u 3 = 0 , which i m p l i e s u,=O, u,=3. I n t h i s case pw, p x , py a r e t h r e e d i s t i n c t l i n e s . Moreover, x and y n e c e s s a r i l y l i e o n t h e s h o r t s i d e , o t h e r w i s e w e h a v e xq > 2 . So, t h e a s s e r t i o n of S t e p 2 i s proved.
.
S t e p 3. The case x,=3 i m p l i e s t h a t G A U { x , y } , where x and y are p o i n t s s a t i s f y one of t h e following conditions:
that
(i) X,YEPV (ii) x ~ u v , (iii) x ~ w p ,
y ~ p x y ~ x v (or x u ) . If x4=3, t h e n x 3 = 7 , x 2 = 6 , x , = 5 . Denote by N t h e t h i r d s e c a n t . It r e s u l t s v = N n u v . We h a v e two p o s s i b i l i t i e s : L,M and N a r e c o n c u r r e n t i n p o r n o t . F i r s t , s u p p o s e p = LflMflN.The p o i n t s o f C are a l l on L U M U N , so w , x , y ~ ? i s i n c e t h e y a r e n o t i n LUM. So C = A U { x , y ) w i t h c o n d i t i o n ( i ) . Next, s u p p o s e p = L n M g N . The p o i n t s o f C are i n LUMUN. P u t y : = C n ( N \ ( L U M ) ) - {w} and x :=C - ( A u ( y ) ) . Denote by u i t h e number of l i n e s which are i - s e c a n t o i C and c o n t a i n p.As i n t h e case of S t e p 2 , w e have: (I)
u3=ul=0
,
u,=3
(11) u3=u2 = u , = 1 .
or
The case ( I ) i m p l i e s a c o n t r a d i c t i o n . ( S i n c e i n AU{y} t h e l i n e s pw and py a r e 2 - s e c a n t o f C, t h e p o i n t x l i e s on t h e f i f t h l i n e t h r o u g h p. I n a n y case xw or xy i s a n u l t e r i o r 4 - s e c a n t ) . F i n a l l y , s u p p o s e t h a t (11) h o l d s . S i n c e i n A U { y } , t h e l i n e s pw and py are 2 - s e c a n t o f C, t h e p o i n t x l i e s on o n e of them. I f X E py, n e c e s s a r i l y x E uv; o t h e r w i s e xw i s a 4 - s e c a n t . T h i s i s t h e case (ii). I f x E pw, t h e l i n e xy n e c e s s a r i l y c o n t a i n s e i t h e r u or v , o t h e r w i s e xy s h o u l d b e a & s e c a n t . T h i s i s t h e case (iii),So, t h e proof of S t e p 3 is f i n i s h e d . S t e p 4. The case x,=4 i m p l i e s t h a t C = A U { x , y } , where x , y s a t i s f y one of t h e following c o n d i t i o n s :
(1)
PEXY
(2)
UEXU
(3) (4)
, WBXY
,
X,Y
(or v ~ x y )
,
XEUV
, w,pexy
;
xEwp
,
points
which
B uv ;
x,y$A
u,v$Xy#pw
are
,
;
yeuv.
Suppose x 4 = 0 and d e n o t e by u i t h e number of i - s e c a n t c o n t a i n i n g p. As i n S t e p s 2 and 3, w e h a v e e i t h e r ( a j u l = u 3 = O = u , or (b) u,=u,=u,=l. I n t h e case ( a ) t h e l i n e s pw, px, py,L,M are a l l d i s t i n c t . I t i s t r i v i a l t h a t e i t h e r xy c o n t a i n s u o r x ~ u v ;o t h e r w i s e x 4 > 4 . So, w e a r e i n s i t u a t i o n s ( 2 ) and ( 3 ) f o r x and y. w i t h y on I n t h e case ( b ) w e n e c e s s a r i l y h a v e e i t h e r p c x y # pw o r x ~ p w o n e o f two r e m a i n i n g l i n e s t h r o u g h p. Moreover, yx c o n t a i n s n e i t h e r u n o r v and y e u v . T h i s c o r r e s p o n d s t o s i t u a t i o n s (1) and ( 4 ) f o r x and y. So, S t e p 4 i s proved. S t e p 5. The case x 4 = 5 i s i m p o s s i b l e .
L. Berardi and F. Eugeni
48
I f x,=5,we h a v e x , = l , x z = 1 2 , x , = 3 . L e t R b e t h e o n l y 3 - s e c a n t , t a k e h E RnC. I f u , a r e t h e number o f i - s e c a n t c o n t a i n i n g h,we have: u,+u,+l+u,=5,uz+2+3u, =9 and r h e n u, =2u, -3 , uz =7-3u,, from which u,=2, uz =u, =1. C o n s e q u e n t l y , e a c h o f t h r e e p o i n t s o n C n R i s i n c i d e n t w i t h e x a c t l y two 4 - s e c a n t . Then x, 2 6 , a c o n t r a d i c t i o n . S t e p 6. As a l a s t r e m a r k , w e n o t e t h a t f o r A U { x , y } e v e r y case i s p r e s e n t e d i n t h e a b o v e s t e p s , e x c e p t t h e f o l l o w i n g : xy # u v , w ~ x y ,p @xy. Note t h a t i n t h i s e x c e p t i o n s xy i s e n t i r e l y c o n t a i n e d i n A U { x , y } which, i n t h i s case, i s n o t a b l o c k i n g s e t . S i n c e t h i s case i s e x c l u d e d by t h e h y p o t h e s i s o f t h e t h e o r e m , t h e a s s e r t i o n 2.4 i s c o m p l e t e l y proved.
3. FINAL REMARKS I n t h i s S e c t i o n w e d e a l about o t h e r q u e s t i o n s connected with blocking sets. We r e c a l l t h a t a n n - f o l d b l o c k i n g s e t i n a n i n c i d e n c e s t r u c t u r e i s t h e u n i o n o f n d i s j o i n t b l o c k i n g sets. I t i s w e l l known ( c f . [ll] ) t h a t i n PG(2,q), q a s q u a r e , t h e r e e x i s t s a Baer p a r t i t i o n i n q-Sq+l Baer s u b p l a n e s . Such a p a r t i t i o n , w h e n q = 4 , i s a 3 - f o l d b l o c k i n g s e t t h a t w e c a l l a Fano p a r t i t i o n . 3.1 REMARK. Denote by 58 a n n - f o l d b l o c k i n g s e t i n PG(2,4), t h e n 1113. Namely: ( a ) n=2, and 9 i s t h e u n i o n e i t h e r o f one b l o c k i n g s e t a n d i t s complement o r o f two b l o c k i n g sets c o n t a i n e d i n e a c h o f them. ( b ) n=3, and 9 i s a Fano p a r t i t i o n .
Another matter i s t h e o p e n problem o f t h e e x i s t e n c e o f b l o c k i n g s e t s i n p r o j e c t i v e o r a f f i n e s p a c e s o f small o r d e r . R e c e n t l y Rajola [ 2 5 ] h a s g i v e n a n example o f b l o c k i n g s e t i n PG(3,5). S o , when q 1 5 , t h e problem i s open i n PG(r.5) ( 1 - 2 4 ) . AG(r.5) ( r 2 3 , c f . [16]) and i n P G ( r , 4 ) (1-23). I n AG(r.4) ( 1 - 2 3 ) no b l o c k i n g set t h e r e e x i s t s . We hope t h a t t h e c l a s s i f i c a t i o n o f b l o c k i n g s e t s i n PG(2.4) c o u l d be u s e f u l f o r t h i s problem. Moreover w e aim t o g i v e a c o m p l e t e c l a s s i f i c a t i o n o f b l o c k i n g sets i n p r o j e c t i v e or a f f i n e s p a c e s o f o r d e r 9 1 9 . S e e t h e l a s t p a r t o f [19] i n t h e s e P r o c e e d i n g s . The l a s t problem t h a t w e want t o r e m a r k i n t h i s S e c t i o n i s t h e problem o f t h e i n d e x ( c f . [ 221, [24] ).Denote by B a b l o c k i n g s e t i n a n i n c i d e n c e s t r u c t u r e . The i n d e x i = i ( B ) of B i s t h e smallest number of b l o c k s s u c h t h a t t h e u n i o n of them c o n t a i n s B. I n [22], i t h a s b e e n proved t h a t i n a p r o j e c t i v e of o r d e r q p l a n e w e h a v e 3 l i ( B ) l q + l f o r a n y b l o c k i n g s e t . I n P G ( 2 , 4 ) , w i t h r e g a r d t o 1 . 4 Theorem o f S e c t i o n 1, w e h a v e t h e f o l l o w i n g s i t u a t i o n . ( a ) A Fano s u b p l a n e F, s e t F u { x } w i t h x on a 3 - s e c a n t , a t r i a n g l e A , a h e r m i t i a n a r c H and t h e b l o c k i n g s e t s o f t y p e e i t h e r A u { x } o r A u { x , y } w i t h x,y w c o l l i n e a r , have a l l index t h r e e . ( b ) E i t h e r F G ( x ) w i t h x o u t s i d e o n e 3 - s e c a n t or A u { x , y ) w i t h x,y,w non- c o l l i n e a r have index four.
It r e m a i n s t o c l a s s i f y t h e i n d e x of t h e complements o f a b o v e c o n s i d e r e d b l o c k i n g sets which are r e d u c i b l e . T h i s i s a t r i v i a l e x e r c i s e , w e remark o n l y t h a t o n e o f them, namely t h e complement of t h e Fano s u b p l a n e h a s i n d e x 5 n e c e s s a r i l y , soeach i n d e x i s p o s s i b l e . REFERENCES
[ 13
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