Blocking Sets in the complement of hyperplane arrangements in projective space

Blo c king Sets in the complemen t of h yp erplane arrangemen ts in pro jectiv e space S. Settepanella ∗ Jan uary 200 8 Abstract It is w ell kn o w that the theory of minimal blo cking sets is studied by sever al aut h or. Another theory whic h is also studied by a large num b er of researc h ers is t he theory of h y p erplane arrangements. W e can remark that the aﬃne space AG ( n, q ) is the complemen t of the line at inﬁnity in P G ( n, q ). Then AG ( n, q ) can b e regarded as the complement of an hyperplane arrangement in P G ( n, q )! Therefore the study of b locking sets in the aﬃne space AG ( n, q ) is simply the study of blo cking sets in the complement of a ﬁn ite arrangemen t in P G ( n, q ). In this paper the author generalizes this remark starting to stu dy the p roblem of existence of blo cking sets in the complement of a given hyp erplane arrangement in P G ( n, q ). As an example she solves the problem for the case of braid arrangemen t. Moreo ver she poses signiﬁcan t questions on this new and interes t in g problem. 1 In tro d uction Throughout this pap er, P G ( n, q ) and AG ( n, q ) will res p ectively denote the n- dimensional pro jective and a ﬃne space over the ﬁnite ﬁeld GF ( q ). A t-blo cking set B in P G ( n, q ) (or AG ( n, q )) is a set B of p oints such that any ( n − t )-dimensional subspace intersects B . A 1-blo cking set is simply called a blo cking set . A t -blo cking set is called minimal , if none of its prop er subs e ts is also a t -blo cking set. The smallest t - blo cking sets hav e b e en characterized by Bo se a nd Bur ton [2]. They proved that the smallest t -blo cking sets in P G ( n, q ) are subspace of dimension t . An old r esult of Bruen [3] states that the second s ma llest minimal blo cking set in the plane P G (2 , √ q ), q a squar e , is (the p oint set) o f a Baer subplane P G (2 , √ q ). There are several survey pap er s ab out blo cking sets (see Sz˜ onyi, G´ acs, W einer [6] and c ha pter 13 of the second editio n of Hirs chf e ld’s b o ok [7 ]). Most of the surveys concentrate on small minimal blo cking sets. But there are also several results ab out lar ge minimal blo c k ing sets. The ﬁr st such result is due to Bruen and Thas [4]. More results on the spectrum of minimal blocking sets in planes of small order can be found in Cossidente, G´ acs et alt [5] Innamorati [8],Inamorati and Maturo [9]. ∗ Dipartiment o di Scienze della Com unicazione,Colle Parco, 641 00 T eramo, Italy 1 Another interesting problem is the existence of minimal Blo cking sets in P G ( n, q ) and AG ( n, q ) for ﬁxed n and q (s e e Mazzo cca-T allini [10] and Beutelspacher- Eugeni [1]). It is w ell know that the theory of minimal blo cking sets is studied b y several author. There is another theory which is also studied by a large num b er of resear chers: the theor y of hyperpla ne arrang ements (see the Orlik-T erao’s bo ok [11] for a survey o n the theor y). An hyperpla ne a rrang e ment is a set o f hyper plane in a given space. The author r emarks that the aﬃne space AG ( n, q ) is the complement of the line at inﬁnity in P G ( n, q ). Then AG ( n, q ) ca n b e regarded a s the complement of a n h yp er plane arr angement in P G ( n, q )! Therefore the study o f blo cking se ts in the aﬃne space AG ( n, q ) is simply the study o f blocking sets in the complement of a ﬁnite a rrang ement in P G ( n, q ). The author generalized this remar k studying the problem of the existence of blo cking sets in the complement of a given hyperplane arrangement in P G ( n, q ). In this pap er a uthor intro duces the ﬁrst results on this new theory , she generalizes so me results of the old theory to the new one. Moreov er she solves the problem o f existence o f blo cking sets in the c omple- men t of a well-known and studied arrang ement: the br aid arr angement , the set of r e ﬂection hyperplane s g enerated by the sy mmetr ic gro up. She concludes the pap er pos ing several questions: 1. on the existence of blo cking sets on the complement of hyperplane a r - rangements; 2. on the c ha racteriza tion of h yp er plane arr angements in P G ( n, q ) which give rise to in teres ting new pro blem on blocking sets; 3. on the link b etw een the tw o theor ies. Clearly also the problem o f small and large blocking sets ca n be s tudied again in the complement of a given hyper plane arra ngement. The author is still working to this problem. Plainly in this pap er author writes blocking s ets instead of min imal blocking sets. 2 General results on Blo c king sets Let P G ( n, q ) ( AG ( n, q )) the n-dimensional pro jective (aﬃne) space o ver a ﬁnite ﬁeld GF ( q ). W e hav e the following: Prop ositi on 1 L et C b e a blo cking set in P G ( n, q ) ( AG ( n, q ) ) with n > 2 and S d a s u bsp ac e of dimension d > 1 , then the interse ction C ∩ S d is a blo cking set in S d . Prop ositi on 2 L et S n − 1 an hyp erplane in P G ( n, q ) ( n > 2 ). If C 1 is a blo cking set in AG ( n, q ) = P G ( n, q ) \ S r − 1 and C 2 is a blo cking set in S r − 1 , then C 1 ∪ C 2 is a blo cking set in P G ( n, q ) . Corollary 1 If P G ( n, q ) and AG ( n +1 , q ) c ontain blo cking s et s then also PG ( r + 1 , q ) do es. 2 The pro of o f the ab ove statements is trivial. Mazzo cca and T allini in [10] prov ed the following: Theorem 1 Ther e is a function b p ( t, q ) ( b a ( t, q ) ), dep ending on t and q, such that P G ( n, q ) ( AG ( n, q ) ) c ontains t-blo cking sets if and only if r ≤ b p ( t, q ) ( r ≤ b a ( t, q ) ). Another interesting genera l r esult on exis tence of blocking set is the follow- ing: Theorem 2 (A. Beutelsp acher-F. Eugeni [1 ]). L et P G ( n, q ) and AG ( n, q ) b e the pr oje ctive and aﬃne n-dimensional sp ac e over a ﬁnite ﬁeld GF ( q ) . If q ≥ 2 n then AG ( n, q ) and P G ( n, q ) c ontains t-blo cking s et s. 3 Blo cki n g sets on the complemen t of h yp er- plane arrangemen ts in PG (n,q) In this sectio n we g eneralize some interesting results on Blo c king Sets in P G ( n, q ) to the c omplement o f hyperpla ne a rrang ement s. Let A = { H 1 , . . . , H m } b e an arr angement of hyp erplanes in P G ( n, q ) and M ( A ) = P G ( n, q ) \ ∪ i =1 ,...m H i be the c omplement of the arrangement. Let A a = { H 1 , . . . , H m } b e an arra ngement in AG ( n, q ) and M a ( A ) = AG ( n, q ) \ ∪ i =1 ,...m H i be the co mplement in the aﬃne s pace. Given an a rrang ement A in P G ( n, q ) ( AG ( n, q )), the c orr esp onding arr ange- ment in P G ( k , q ) ( AG ( k, q )) for k 6 = n is the arra ngement given b y hyperplanes with the same equatio ns of thos e in A . F rom now on we will a lso use A to r efer to corr esp onding arr angements of A . Let us r emarks that the statemen ts of pro p ositions 1, 2 and co r ollary 1 a b ov e hold trivially also for Blo cking sets in the complements M ( A ) ⊂ P G ( n, q ) and M a ( A ) ⊂ AG ( n, q ). Moreov er we hav e the g eneralizatio n o f Mazzo cca-T allini’s theor em: Theorem 3 L et M ( A ) b e the c omplement of an arr angement A in the pr oje c- tive (aﬃne) sp ac e P G ( n, q ) ( AG ( n, q ) ). Then we c an ﬁnd a function b p, A ( t, q ) ( b a, A ( t, q ) ), dep ending on t and q, such t hat M ( A ) ⊂ P G ( n, q ) ( M ( A ) ⊂ AG ( n, q ) ) has t-blo cking sets if and only if r ≤ b p, A ( t, q ) ( r ≤ b a, A ( t, q ) ). Pro of Let d M ( A ( n,q )) be the maxim um of the dimensions of linear subspaces in M ( A ( n, q )). Then, by simple geometr ic considera tion, we remar k that { d M ( A ( n,q )) : n ∈ N } is an increas ing function. Let us suppo se tha t { r n : n ∈ N } is a sequence of in teg er suc h that for all r n there is an h - blo c k ing set B ( r n ) in M ( A ( n, q )). Then fo r d M ( A ( r n ,q )) > h let B ( r n ) ∩ S ( r n ) b e the intersection b etw een B ( r n ) and a linear v a riety S ( r n ) in M ( A ( r n , q )) of dimension d M ( A ( r n ,q )) . Then 3 B ( r n ) ∩ S ( r n ) is an h -blo cking set in S ( r n ), but this co ntradicts theor em 1.  The braid arrangement Let us consider the Br aid arr angement A ( A n,q ) = { H i,j } 1 ≤ i

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