Combinatorics of discriminantal arrangements
CCOMBINATORICS OF DISCRIMINANTAL ARRANGEMENTS
SIMONA SETTEPANELLA AND SO YAMAGATAA bstract . In 1985 Crapo introduced in [4] a new mathematical object that he called geometry of circuits . Four years later, in1989, Manin and Schechtman defined in [14] the same object and called it discriminantal arrangement , the name by which itis known now a days. Subsequently in 1997 Bayer and Brandt ( see [3] ) distinguished two di ff erent type of those arrangementscalling very generic the ones having intersection lattice of maximum cardinality and non very generic the others. Results onthe combinatorics of very generic arrangements already appear in Crapo [4] and in 1997 in Athanasiadis [2] while the firstknown result on non very generic case is due to Libgober and the first author in 2018. In their paper [13], they studied thecombinatorics of non very generic case in rank 2 . In this paper we further develop their result providing a su ffi cient conditionfor the discriminantal arrangement to be non very generic which holds in rank r ≥
1. I ntroduction
In 1989, Manin and Schechtman ([14]) introduced a family of arrangements of hyperplanes generalizing clas-sical braid arrangements, which they called the discriminantal arrangements (p.209 [14]). Such an arrangement B ( n , k , A ) , n , k ∈ N for k ≥ A = { H , ..., H n } of a collection of hyperplanes in generalposition in C k , i.e., such that dim (cid:84) i ∈ K , | K | = k H i =
0. It consists of parallel translates of H t , ..., H t n n , ( t , ..., t n ) ∈ C n which fail to form a general position arrangement in C k . B ( n , k , A ) can be viewed as a generalization of the braidarrangement ([16]) with which B ( n , = B ( n , , A ) coincides.These arrangements have several beautiful relations with diverse problems such as the Zamolodchikov equation withits relation to higher category theory (see Kapranov-Voevodsky [10], see also [8],[9]), the vanishing of cohomologyof bundles on toric varieties ([17]), the representations of higher braid groups (see [11]) and, naturally, with combina-torics. The latter is the connection we are mainly interested in and it goes from matroids to special configurations ofpoints, from fiber polytopes to higher Bruhat orders.Manin and Schechtman introduced discriminantal arrangements as higher braid arrangements in order to introducehigher Bruhat orders which model the set of minimal path through a discriminantal arrangement. Even if Zieglershowed ( see Theorem 4.1 in [20] ) in 1991 that we have to choose a cyclic arrangement instead of discriminantal ar-rangement for this, few years later, in a subsequent work (see [7]) Felsner and Ziegler reintroduced the combinatoricsof discriminantal arrangement in the study of higher Bruhat orders (this connection uses fiber polytopes as observed byFalk in [6]). From a di ff erent perspective, unknown in the literature of discriminantal arrangement until Athanasiadispointed it out in 1999 ( see [2]), Crapo introduced for the first time in 1985 (see [4]) what he called geometry of circuits and which is the matroid M ( n , k , C ) of circuits of the configuration C of n generic points in R k . The circuits of thematroid M ( n , k , C ) are the hyperplanes of B ( n , k , A ) , A arrangement of n hyperplanes in R k orthogonal to the vectorsjoining the origin with the n points in C ( for further development see [5] ).Both Manin-Schechtman ([14]) and Crapo ([4]) were mainly interested in arrangements B ( n , k , A ) for which the in-tersection lattice is constant when A varies within a Zariski open set Z in the space of general position arrangements.Crapo shows that, in this case, the matroid M ( n , k ) is isomorphic to the Dilworth completion of the k -th lower trunca-tion of the Boolean algebra of rank n . More recently in [2], Athanasiadis proved a conjecture by Bayer and Brandt ( see[3]) providing a full description of combinatorics of B ( n , k , A ) when A belongs to Z . Following [2] (more preciselyBayer and Brandt ), we call arrangements A in Z very generic , non very generic otherwise. Mathematics Subject Classification.
Key words and phrases.
Discriminantal arrangements, realizable matroids, combinatorics of arrangements.The second author was supported by the Program for Leading Graduate Schools (Hokkaido University Ambitious Leader’s Program) and JSPSResearch Fellowship for Young Scientists Grant Number 20J10012. a r X i v : . [ m a t h . C O ] J a n SIMONA SETTEPANELLA AND SO YAMAGATA
However Manin and Schechtman do not describe the set Z of very generic arrangements explicitly, which, in time, ledto the misunderstanding that the combinatorial type of B ( n , k , A ) was independent from the arrangement A (see forinstance, [15], sect. 8, [16] or [12]). Neither Crapo in [4] provided a description of Z even if he presented the firstknown example of a non very generic arrangement: 6 lines in generic position in R which admit translated that arerespectively sides and diagonals of a quadrilateral as in Figure 1 (Crapo calls it a quadrilateral set). Few years laterin 1994, Falk provided an higher dimensional example of non very generic arrangement of 6 planes in R ( see [6]).Similar to Crapo’s example, Falk’s example too turned out to be related to a special configuration of lines, this time inprojective plane ( see [18],[19]). H H H H H H H H H H H H H H H H H H P P P P F igure
1. Central generic arrangement of 6 lines in R , its generic translation on the left and its non(very) generic translation on the right.In 2018 the first general result on non very generic arrangements is provided. In [13] Libgober and the first authordescribed a su ffi cient geometric condition on the arrangement A to be non very generic. This condition ensures that B ( n , k , A ) admits codimension 2 strata of multiplicity 3 which do not exist in very generic case. It is given in termsof the notion of dependency for the arrangement A ∞ in P k − of hyperplanes H ∞ , , ... H ∞ , n which are the intersections ofprojective closures of H , ..., H n ∈ A with the hyperplane at infinity. Their main result shows that B ( n , k , A ) , k > A ∞ is an arrangement in P k − admitting a restriction which is a dependent arrangement. This construction generalizes Falk’s example which corresponds to the case n = , k = A of 6 planes in R (resp. C ) for which the rank 2 Here restriction is the standard restriction of arrangements to subspaces as defined in [16].
OMBINATORICS OF DISCRIMINANTAL ARRANGEMENTS 3 intersections of B (6 , , A ) are in minimal number corresponds to Pappus’s ( resp. Hesse’s ) configuration providinga main example of what conjectured by Crapo that the intersection lattice of discriminantal arrangement represents acombinatorial way to encode special configurations of points in the space. Notice that in [18] the authors connectedthe non very generic arrangements A of n planes in C to well defined hypersurfaces in Grassmannian Gr (3 , n ).In this paper we advance the study of non very generic arrangements and generalize the dependency condition given in[13] providing a su ffi cient condition for the existence in rank r ≥ B ( n , k , A ) , A ∈ Z . In particular we call simple an intersection of r hyperplanes in B ( n , k , A )which satisfies the property that if the arrangement A is very generic then all simple intersections of multiplicity r have rank r ( that is they are r hyperplanes intersecting transversally). Then we provide both geometric and algebraicnecessary and su ffi cient conditions for existence of simple intersections of multiplicity r in rank strictly lower than r ,i.e. simple non very generic intersections. This result firstly connect configurations of non very generic points to specialfamilies of graphs ( called K T -configurations ) which help to understand B ( n , k , A ) for A (cid:60) Z (as conjectured byCrapo in [4]). Secondly it reduces the geometric problem of the existence of special ( non very generic) configurationsof points to a combinatorial problem on the numerical properties that r subsets of indices L i ⊂ { , . . . , n } , i = , . . . , r of cardinality k + K T -configuration, T = { L , . . . , L r } , to give rise to a simple non verygeneric intersection. The latter problem is left open together with the problem of necessary and su ffi cient conditionsfor existence of intersections in B ( n , k , A ) which are nor simple nor very generic.The content of the paper is the following. In Section 2, we recall the definition of discriminantal arrangement and basicproperties of the intersection lattice of discriminantal arrangement in very generic case. We also give the definition of simple intersection. In Section 3, we introduce the notion of K T -translated and K T -configuration associated to a genericarrangement A providing a geometric condition for A to be non very generic ( Theorem 3.9 ). Finally we define the K T -vector condition. In Section 4, we prove that the existence of a finite number of sets of vectors which satisfy the K T -vector condition is su ffi cient condition for A to be non very generic (Theorem 4.5). In the last section we provideexamples of non very generic arrangements obtained by imposing the condition stated in Theorem 4.5.2. P reliminaries Discriminantal arrangement.
Let H i , i = , ..., n be a central arrangement in C k , k < n which is generic , i.e.any m hyperplanes intersect in codimension m at any point except for the origin for any m ≤ k . We will call such anarrangement a central generic arrangement . Space of parallel translates S ( H , ..., H n ) (or simply S when dependenceon H i is clear or not essential) is the space of n -tuples of translates H , . . . , H n such that either H i ∩ H i = ∅ or H i = H i for any i = , . . . , n .One can identify S with n -dimensional a ffi ne space C n in such a way that ( H , . . . , H n ) corresponds to the origin. Inparticular, an ordering of hyperplanes in A determines the coordinate system in S (see [13]).Given a central generic arrangement A in C k formed by hyperplanes H i , i = , . . . , n the trace at infinity , denoted by A ∞ , is the arrangement formed by hyperplanes H ∞ , i = ¯ H i ∩ H ∞ in the space H ∞ (cid:39) P k − ( C ), where ¯ H i are projectiveclosures of a ffi ne hyperplanes H i in the compactification P k ( C ) of C k (cid:39) P k ( C ) \ H ∞ . Notice that condition of genericityis equivalent to (cid:83) i H ∞ , i being a normal crossing divisor in P k − ( C ), i.e. A ∞ is a generic arrangement.The trace A ∞ of an arrangement A determines the space of parallel translates S (as a subspace in the space of n -tuples of hyperplanes in P k ). Fixed a generic central arrangement A , consider the closed subset of S formed by thosecollections which fail to form a generic arrangement. This subset of S is a union of hyperplanes D L ⊂ S (see [14]).Each hyperplane D L corresponds to a subset L = { i , . . . , i k + } ⊂ [ n ] (cid:66) { , . . . , n } and it consists of n -tuples of translatesof hyperplanes H , . . . , H n in which translates of H i , . . . , H i k + fail to form a generic arrangement. The arrangement B ( n , k , A ) of hyperplanes D L is called discriminantal arrangement and has been introduced by Manin and Schechtmanin [14] . Notice that B ( n , k , A ) depends on the trace at infinity A ∞ hence it is sometimes more properly denoted by B ( n , k , A ∞ ). Notice that, in general, generic, referred to an arrangement of hyperplanes, has a slightly di ff erent meaning. With an abuse of notation, we usethe word generic in this case since the defined property is equivalent to the existence of a translated of the given central arrangement which is genericin the classical sense. Notice that Manin and Schechtman defined the discriminantal arrangement starting from a generic arrangement instead of its central translatedas we do in this paper. For our purpose the latter is a more convenient choice.
SIMONA SETTEPANELLA AND SO YAMAGATA
Very generic and non very generic discriminantal arrangements.
It is well known (see, among others [4],[14])that there exists an open Zarisky set Z in the space of (central) generic arrangements of n hyperplanes in C k , such thatthe intersection lattice of the discriminantal arrangement B ( n , k , A ) is independent from the choice of the arrangement A ∈ Z . Bayer and Brandt in [3] call the arrangements
A ∈ Z very generic and the ones which are not in Z , non verygeneric . We will use their terminology in the rest of this paper. The name very generic comes from the fact that in thiscase the cardinality of the intersection lattice of B ( n , k , A ) is the largest possible for any (central) generic arrangement A of n hyperplanes in C k .In [4] Crapo proved that the intersection lattice of B ( n , k , A ) , A ∈ Z is isomorphic to the Dilworth completion of the k -times lower-truncated Boolean algebra B n ( see Theorem 2. page 149 ). A more precise description of this latticeis due to Athanasiadis who proved in [2] a conjecture by Bayer and Brandt which stated that the intersection latticeof the discriminantal arrangement in very generic case is isomorphic to the collection of all sets { S , . . . , S m } , S i ⊂ [ n ] = { , . . . , n } , | S i |≥ k +
1, such that(1) | (cid:91) i ∈ I S i | > k + (cid:88) i ∈ I ( | S i | − k ) for all I ⊂ [ m ] = { , . . . , m } , | I |≥ . The isomorphism is the natural one which associate to the set S i the space D S i = (cid:84) L ⊂ S i , | L | = k + D L , D L ∈ B ( n , k , A ) ofall translated of A having hyperplanes indexed in S i intersecting in a not empty space. In particular { S , . . . , S m } willcorrespond to the intersection (cid:84) mi = D S i .If A is very generic and the condition in equation (1) is satisfied, this implies that the subspaces D S i , i = , . . . , m intersect transversally (Corollary 3.6 in [2]) or, equivalently, since rank D S i = | S i | − k , that(2) rank m (cid:92) i = D S i = m (cid:88) i = ( | S i | − k )Notice that the condition in equation (1) is satisfied (see also [2]) if(3) (cid:92) i ∈ I D S i (cid:44) D S , | S | > k + I ⊂ [ r ] = { , . . . , r } , | I |≥ . The fact that for all sets { S , . . . , S m } which satisfy condition in equation (3) the condition in equation (2) is alsosatisfied corresponds to the definition provided by Crapo in [4] for discriminantal arrangement in very generic case(which he called geometry of circuits ). From those considerations we can get the following remark. Remark 2.1.
Let A (cid:48) be a translated of a central generic arrangement A such that the hyperplanes in A (cid:48) indexed insubsets L i ⊂ [ n ] , | L i | = k + , i = , . . . , r intersect at the points P i = (cid:84) p ∈ L i H p (cid:44) ∅ and satisfy (cid:84) p ∈ L i ∪{ t } H p = ∅ for anyt (cid:60) L i . Then A (cid:48) is an element in the intersection (cid:84) ri = D L i and A (cid:48) (cid:60) D S i for any S i ⊃ L i , i.e. (cid:84) i ∈ I D L i (cid:44) D S , | S | > k + for any I ⊂ [ r ] = { , . . . , r } , | I |≥ . In particular if A is very generic then, by equation (2), D L i are r hyperplanesintersecting transversally, i.e. rank (cid:84) ri = D L i = r. Contrary to the very generic case, very few is known about the non very generic case. In recent papers the first author(see [13]) and the first and second authors (see [18], [19]) showed that non very generic arrangements are arrangementswhich hyperplanes give rise to special configurations ( e.g. Pappus’s configuration or Hesse configuration). Followingthis direction, in the rest of the paper we further develop the result in [13] providing a geometric and an algebraiccondition for a central generic arrangement A to be non very generic. In order to do that we will use Remark 2.1which essentially states that if A is a central generic arrangement of n hyperplanes in C k for which there exist r subsets L i ⊂ [ n ], | L i | = k + , i = , . . . , r such that(1) for any translated A (cid:48) ∈ (cid:84) ri = D L i of A the hyperplanes in A (cid:48) satisfy the additional condition (cid:84) p ∈ L i ∪{ t } H p = ∅ for any t (cid:60) L i ,(2) rank (cid:84) ri = D L i < r then A is non very generic. The following definition will be very useful in the rest of the paper. Here Crapo followed the preference of his advisor Rota who rarely used the name matroid.
OMBINATORICS OF DISCRIMINANTAL ARRANGEMENTS 5
Definition 2.2.
An element X in the intersection lattice of the discriminantal arrangement B ( n , k , A ) is said to be simple intersection if X = (cid:84) ri = D L i , | L i | = k + and (cid:84) i ∈ I D L i (cid:44) D S , | S | > k + for any I ⊂ [ r ] , | I |≥ . We callmultiplicity of the simple intersection X the number r of hyperplanes intersecting in X. By the above considerations we can conclude that if the intersection lattice of the discriminantal arrangement B ( n , k , A )contains a simple intersection of rank strictly less than its multiplicity, then A is non very generic. This will play animportant role in the rest of the paper since we will focus on a necessary and su ffi cient condition for the existence ofsuch a simple intersection.2.3. Motivating examples.
In this subsection we provide the two main examples given by Crapo (see [4]) and Falk(see [5] and [6]) of simple but non very generic intersections. Those two examples inspired the rest of the content ofthis paper.
Crapo example is illustrated in Figure 1. In this case the arrangement A t on the right of Figure 1 is an element in thesimple intersection X = (cid:84) i = D L i with L = { , , } , L = { , , } , L = { , , } , L = { , , } . On the other hand, theonly rank 4 intersection of B (6 , , A ) is given by D [6] , the space of all central translated of A . Since A t is not central,this implies that X (cid:44) D [6] and hence rank X <
4, that is X is a simple intersection of multiplicity 4 and rank 3 <
4. Thatis the arrangement in Figure 1 is non very generic.
Falk example is illustrated in Figure 2. Let B (6 , , A ∞ ) be the discriminantal arrangement associated to a genericarrangement A of 6 hyperplanes H i in R which satisfy the condition that H i ∩ H i + ∩ H ∞ , i = , , A admits a translation A (cid:48) which belongsto the simple intersection X = (cid:84) i = D L i with L = { , , , } , L = { , , , } , L = { , , , } , that is, in particular, A (cid:48) is not a central arrangement. On the other hand the only element in rank 3 in B (6 , , A ) is D [6] the space of all centraltranslated of A hence rank X <
3, that is X is a simple intersection of multiplicity 3 and rank 2 <
3, i.e. A is non verygeneric. F igure
2. Figure of non very generic arrangement with 6 hyperplanes in R . SIMONA SETTEPANELLA AND SO YAMAGATA
3. A geometric condition for non very genericity
In this section we provide a necessary and su ffi cient condition on a central generic arrangement A in C k for theexistence of a simple intersection X of multiplicity r and rank X < r in the intersection lattice of the discriminantalarrangement B ( n , k , A ). As noticed in previous section, this is a su ffi cient condition for A to be non very generic. Notation 3.1.
To begin with let us fix some notations we will use throughout the paper. • A is a central generic arrangement of n hyperplanes in C k • For each subset L of { , . . . , n } with | L | = k + , D L ⊂ C n will denote the hyperplane in B ( n , k , A ) correspond-ing to the subset L. • Fixed a set T = { L , . . . , L r } of subsets L i ⊂ [ n ] , | L i | = k + , for any arrangement A = { H , . . . , H n } translatedof A we will denote by P i = (cid:84) p ∈ L i H p and H i , j = (cid:84) p ∈ L i ∩ L j H p . Notice that P i is a point if and only if A ∈ D L i ,it is empty otherwise. K T -translated and K T -configurations. Let A = { H , . . . , H n } be a central generic arrangement in C k , T = { L , . . . , L r } fixed as in Notation 3.1 and such that the conditions(4) r (cid:91) i = L i = (cid:91) i ∈ I ⊂ [ r ] , | I | = r − L i and L i ∩ L j (cid:44) ∅ are satisfied for any subset I ⊂ [ r ] , | I | = r − ≤ i < j ≤ r . In the rest of the paper a set T whichsatisfies those properties will be called an r - set .A translated A = { H , . . . , H n } of A will be called K T or K T -translated if (cid:84) p ∈ L i H p (cid:44) ∅ and (cid:84) p ∈ L i ∪{ t } H p = ∅ for any i ∈ [ r ] and t (cid:60) L i . The complete graph ( as depicted in Figure 3) having the points P i = (cid:84) p ∈ L i H p as vertices and thevectors P i P j joining P i and P j as edges will be called K T -configuration and denoted by K T ( A ) ( examples of graphs K T ( A ) for | T | = , , P i P j ∈ H i , j = (cid:84) p ∈ L i ∩ L j H p (cid:44) ∅ for any 1 ≤ i < j ≤ r . A will be called almost- K T if it is K T but for one hyperplane H l and a set S l , i.e. if there exists a hyperplane H l ∈ A , l ∈ (cid:83) ri = L i \ (cid:84) ri = L i , and S l ⊆ { L i ∈ T | l ∈ L i } such that (cid:84) p ∈ L i H p = ∅ for any L i ∈ S l and (cid:84) p ∈ L j H p (cid:44) ∅ for any L j ∈ T \ S l . Notice that since A is a central generic arrangement and | L i | = k +
1, then (cid:84) p ∈ L i \{ l } H p (cid:44) ∅ for any L i ∈ S l . Moreover by condition in equation (4) if the set { L i ∈ T | l ∈ L i } is not empty then its cardinality is | { L i ∈ T | l ∈ L i } |≥
2. If we keep the notation P i = (cid:84) p ∈ L i \{ l } H p , L i ∈ S l , P i = (cid:84) p ∈ L i H p , L i (cid:60) S l , the complete graphhaving P i as vertices and P i P j as edges will be called almost K T -configuration and denoted by K T \ S l ( A ). P P P . . . . . . P r ... H , H , r H , F igure K T -configuration for | T | = r OMBINATORICS OF DISCRIMINANTAL ARRANGEMENTS 7 P P P H , H , H , P P P P H , H , H , H , H , H , P P P P P H , H , H , H , H , H , H , H , H , H , F igure
4. Left: | T | =
3, Center: | T | =
4, Right: | T | = Remark 3.2.
Notice that as soon as L i ∩ L j (cid:44) ∅ then H i , j = (cid:84) p ∈ L i ∩ L j H p (cid:44) ∅ for any translated A of A . Indeed H i , j is an a ffi ne space having as underlining vector space H i , j = (cid:84) p ∈ L i ∩ L j H p of codimension the cardinality | L i ∩ L j | ofL i ∩ L j . In particular for any translated A such that P i = (cid:84) p ∈ L i H p (cid:44) ∅ there is one and only one vector v i , j ∈ H i , j suchthat v i , j applied to the point P i has exactly P j as ending point. This is very relevant fact as this essentially provide, innon very generic case, what Gale called a ffi ne dependency. Remark 3.3.
Similarly to the K T -configuration we could define the ∆ T -configuration as the simplicial complex havingas t-face P i . . . P i t + ∈ (cid:84) p ∈∩ t + j = L ij H p (cid:44) ∅ . Notice that in general, the intersection (cid:84) p ∈∩ t + j = L ij H p can be empty, that is ∆ T is not a simplex. As pointed out by Crapo in [4] , this simplicial complex may play a fundamental role in the study ofnon very generic arrangements. With the notations introduced above, we provide the following main definition.
Definition 3.4.
A central generic arrangement A of n hyperplanes in C k is called (r,s)-dependent if there exist anr-set T = { L , . . . , L r } , an index l ∈ (cid:83) ri = L i \ (cid:84) ri = L i and a subset S l ⊆ { L i ∈ T | l ∈ L i } , | S l | = s, such that anyalmost K T -configuration K T \ S l ( A ) gives rise to a K T -configuration K T ( A (cid:48) ) with the K T -translated A (cid:48) obtained fromthe almost K T -translated A by a suitable translation of the hyperplane H l ∈ A . If s = , then we call A r-dependent. Example 3.5 (Crapo’s example) . Let us consider the Crapo’s example in Subsection 2.3. Looking at Figure 1 we caneasily check that if we choose l = , then the almost K T -configuration given by P = H ∩ H ∩ H , P = H ∩ H ,P = H ∩ H ∩ H , P = H ∩ H becomes a K T -configuration by the translation of H such that P ∩ H (cid:44) ∅ (asdepicted in Figure 5). That is the arrangement A depicted in Figure 1 is -dependent. Remark that if A ∈ (cid:84) i = D L i ,then A ∈ D L . The definition of ( r , s )-dependency generalizes the definition of dependency given in [13]. Indeed we have the followingremark which, in particular, applies to Falk’s example ( see Example 3.7 ). Remark 3.6 (Dependency and 3-dependency) . Let’s focus on the case in which T = { L , L , L } is a set of cardinality3 to show that, the definition of r-dependency is a generalization of the definition of dependency given in [13] . In orderto do that it is enough to show that both conditions, i.e. 3-dependency and dependency, are equivalent to the conditionthat the space H i , j is a subspace of H i , k + H k , j . Dependency.
Recall that an arrangement A of s hyperplanes in C s − , s ≥ is dependent if it exists a set T = { L , L , L } of subsets L i ⊂ [3 s ] such that | L i | = s, | L i ∩ L j | = s, | (cid:83) i = L i | = s and spaces H i , j = (cid:84) p ∈ L i ∩ L j H p span asubspace of dimension s − in C s − . The condition | L i ∩ L j | = s implies that H i , j are spaces of codimension s, that isof dimension s − in C s − . Moreover | (cid:83) i = L i | = s implies that the (cid:83) i = L i is disjoint union of the three sets L i ∩ L j ,that is any two subspaces H i , j are in direct sum, i.e. H i , k ⊕ H k , j span a space of dimension s − . Hence dependencycondition is equivalent to the fact that H i , j belongs to the space generated by H i , k ⊕ H k , j . -dependency First of all notice that K T -configuration when | T | = is equivalent to the fact that P i P j = P i P k + P k P j SIMONA SETTEPANELLA AND SO YAMAGATA H H H H H H P P P P H H H H H H P P P P F igure K T \ S ( A ), S = { L , L } on the left and K T ( A (cid:48) ) on the right. (see Figure 4) and that since (cid:83) i = L i = (cid:83) i ∈ I ⊂ [3] , | I | = L i then any index l ∈ (cid:83) i = L i \ (cid:84) i = L i belongs to exactly twodi ff erent subsets L i and L j . The -dependency condition is then equivalent to the fact that any translation for which thevertex P i = (cid:84) p ∈ L i H p (cid:44) ∅ exists then P j = (cid:84) p ∈ L j H p (cid:44) ∅ exists and P i P j = P i P k + P k P j for any P i , P j ∈ H i , j , that isH i , j is a subspace of H i , k + H k , j . In particular, H i , k + H k , j = H i , k ⊕ H k , j are in direct sum as soon as (cid:84) i = L i = ∅ . Example 3.7. [Falk’s example] Consider the Falk’s example in Subection 2.3. In this case A is an arrangement of 6hyperplanes in R and the set T = { L , L , L } is given by L = { , , , } , L = { , , , } and L = { , , , } whichsatisfy the conditions | L i | = , | L i ∩ L j | = and | (cid:83) i = L i | = . Since the spaces H i , j , ∞ = (cid:84) p ∈ L i ∩ L j H p ∩ H ∞ span asubspace of dimension in H ∞ then H i , j span a space of dimension in R , that is A is a dependent arrangement.On the other hand, let’s choose the index l = and S l = { L , L } . Then for any translated A of A such thatP = (cid:84) p ∈ L H p (cid:44) ∅ exists, the translated H (cid:48) of H for which P = H (cid:48) ∩ (cid:84) p ∈ L \{ } H p (cid:44) ∅ also satisfies P = H (cid:48) ∩ (cid:84) p ∈ L \{ } H p (cid:44) ∅ . Moreover P P = P P + P P , that is A is -dependent. Notice that the condition of r -dependency is non trivial one. Indeed by (cid:83) ri = L i = (cid:83) i ∈ I ⊂ [ r ] , | I | = r − L i it follows that anyindex l ∈ (cid:83) ri = L i has to belong to at least two di ff erent subsets L i ’s. Hence if L i (cid:44) L j are two di ff erent subsetscontaining the index l the fact that H l is a translated of H l for which (cid:84) p ∈ L i \{ l } H p ∩ H l (cid:44) ∅ does not imply, in general,that (cid:84) p ∈ L j \{ l } H p ∩ H l (cid:44) ∅ . In particular ( r , s )-dependency is always non trivial for any 1 < s ≤| { L i ∈ T | l ∈ L i } | .It is also a simple remark that with the notations in Definition 3.4 A is ( r , s )-dependent if and only if any translatedarrangement A ∈ (cid:84) L i ∈ T \ S l D L i ∩ D L j , for some L j ∈ S l , satisfies A ∈ (cid:84) L i ∈ T D L i . That is A is ( r , s )-dependent if andonly if there exist an r -set T = { L , . . . , L r } such that the intersection X = (cid:84) L i ∈ T D L i is simple and a subset S l ⊂ T suchthat for any L j ∈ S l (5) (cid:92) L i ∈ T \ S l D L i ∩ D L j = (cid:92) L i ∈ T D L i . In particular, the equality (5) implies that the rank X < r and hence the following Lemma holds. Lemma 3.8.
The discriminantal arrangement B ( n , k , A ) admits a simple intersection X of multiplicity r and rank X < r if and only if A is ( r , s ) -dependent for some s > . An immediate consequence of Lemma 3.8 and the remarks in Subsection 2.2 is the following main theorem.
Theorem 3.9.
If a central generic arrangement A of n hyperplanes in C k is ( r , s ) -dependent for some s > then A is non very generic. OMBINATORICS OF DISCRIMINANTAL ARRANGEMENTS 9
Let’s remark that while Theorem 3.9 provides a geometric condition which gives rise to non very genericity, it isquite hard to verify if an arrangement satisfies such condition. In the rest of the paper we will focus on providing anequivalent condition for ( r , s )-dependency which can be computationally verified and which allows to build non verygeneric arrangements. In order to do this we will need to introduce the following vector condition.Let T be an r -set, D ([ r ]) = { ( i , j ) ∈ Z r × Z r | i + < j } be the set of not adjacent pairs of integers mod r and for any ( i , j )in D ([ r ]), v i , j vectors defined as linear combinations of the form(6) v i , j (cid:66) j − (cid:88) p = i v p , p + = − i − (cid:88) p = j v p , p + , v p , p + ∈ H p , p + where the right summand is intended from j to the first representative h > j such that h ≡ r i − P p + v p , p + , i.e. v p , p + applied to points P p = (cid:84) i ∈ L p H i , H i ’s translated of H i , can be regarded as sides of an r -gon having as vertices the applications points P p and as edges the vectors v p , p + directed counter clockwise as depicted in Figure 6. Definition 3.10.
We say that vectors v i , j satisfy the K T -vector condition if there exist a hyperplane H l ∈ A , l ∈ (cid:83) ri = L i \ (cid:84) ri = L i and a subset S l ⊆ { L i ∈ T | l ∈ L i } such that if v i , j ∈ H i , j = (cid:84) p ∈ L i ∩ L j H p , L i (cid:60) S l , and v i , j ∈ (cid:84) p ∈ L i ∩ L j \{ l } H p , L i , L j ∈ S l , then v i , j ∈ H i , j for any ( i , j ) ∈ D ([ r ]) . By definition if A is a K T -translated of A , then each vector v i , j ∈ H i , j is a vector that applied to the point P i has P j asending point in the K T -configuration K T ( A ), that is P i + v i , j = P j ∈ (cid:92) p ∈ L j H p as described in Figure 6. In particular the vector P i P j is a translated of v i , j and the following lemma holds. P i P i + P i + . . . P j − P j P j + . . . P i − P i − v i , i + v i , j − v i , j + v i + , i + v j − , j v j , j + v i − , i − v i − , i v i , j F igure
6. Diagonal vectors v i , j can be written as a sum of side vectors v p , p + . Lemma 3.11.
A central generic arrangement A of n hyperplanes in C k is ( r , s ) -dependent if and only if it exists a set T = { L , . . . , L r } such that any set of vectors { v i , j } defined as in equation (6) satisfies the K T -vector condition for someH l ∈ A , l ∈ (cid:83) ri = L i \ (cid:84) ri = L i and a subset S l of cardinality s. In the next section we will use the K T -vector condition to simplify the condition for ( r , s )-dependency. In particularwe will show that it is not needed, as stated in Lemma 3.11, that any set of vectors { v i , j } defined as in equation (6)satisfies the K T -vector condition in order for A to be ( r , s )-dependent, but it is su ffi cient that the K T -vector conditionis satisfied by just a finite subset of them.4. A n algebraic condition for non very genericity In this section A t = { H x , . . . , H x n n } will denote the translation of the central generic arrangement A = { H , . . . , H n } in C k by the vector t = ( x , . . . , x n ) ∈ C n , i.e., H x i i = H i + α i x i , α i unitary vector normal to H i . Notice that this notationgives rise to a natural identification of the space S = S [ A ] of parallel translates of A with C n in such a way that thearrangement A corresponds to the origin.The discriminantal arrangement B ( n , k , A ) is not essential arrangement of center D [ n ] = (cid:84) L ⊂ n , | L | = k + D L (cid:39) C k givenby all translated A t of A which are central arrangements. Hence we can consider its essentialization ess ( B ( n , k , A ))in C n − k (cid:39) S / D [ n ] . An element A t ∈ ess ( B ( n , k , A )) will corresponds uniquely to a translation t ∈ C n / C (cid:39) C n − k , C = { t ∈ C n | A t is central } . The following proposition arises naturally. Proposition 4.1.
Let A be a generic central arrangement of n hyperplanes in C k . Translations A t , . . . , A t d of A are linearly independent vectors in S / D [ n ] (cid:39) C n − k if and only if t , . . . , t d are linearly independent vectors in C n / C. Given A t , . . . , A t d K T -translated of A , we will say that the K T -configurations K T ( A t i ) are independent if A t i , i = , . . . , d are.Let’s consider vectors { v i , j } introduced in equation (6) associated to an r -set T . We can remark the following threefacts:(1) To each K T -configuration K T ( A t ) of a translated arrangement A t , t = ( x , . . . , x n ), corresponds a unique family { v ti , j } of vectors such that P ti + v ti , j = P tj = (cid:84) p ∈ L j H x p p . In the rest of the section we will denote by { v ti , j } thefamily of vectors associated to K T ( A t ). Notice that the converse is not uniquely defined since two di ff erent K T -configurations can define the same family { v i , j } .(2) By construction vectors { v ti , j } satisfy the property that v tk , l = v ti . l − v ti , k ( this can be easily seen looking attranslated of vectors v i , j ’s represented in Figure 6). Then the set { v ti , j } is uniquely determined by any subset ofthe form { v tj , i , v ti , j } j (cid:44) i for a fixed index i ∈ [ r ]. For simplicity in the rest of the Section we will use the set { v tj , i , v ti , j } j (cid:44) i instead of { v ti , j } and we will call it K T - vector set .(3) Any family of vectors { v ti , j } associated to a K T -configuration K T ( A t ) satisfies, by construction, the K T -vectorcondition and, consequently, the family of vectors { v ti , j } built via the relations in (2) from the K T -vector set { v tj , i , v ti , j } j (cid:44) i does. Hence it is enough to say that { v tj , i , v ti , j } j (cid:44) i satisfies the K T -vector condition since { v ti , j } satisfies it if and only if { v tj , i , v ti , j } j (cid:44) i does.Given a K T -vector set we can naturally define operation of multiplication by a scalar a { v tj , i , v ti , j } j (cid:44) i (cid:66) { av tj , i , av ti , j } j (cid:44) i , a ∈ C and sum of two di ff erent K T -vector sets { v t j , i , v t i , j } j (cid:44) i + { v t j , i , v t i , j } j (cid:44) i = { v t j , i + v t j , i , v t i , j + v t i , j } j (cid:44) i . With above notations and operations, we have the following definition.
Definition 4.2.
For a fixed set T , d di ff erent K T -vector sets {{ v t h j , i , v t h i , j } j (cid:44) i } h = ,..., d are linearly independent if and onlyif for any a , . . . , a d ∈ C such that (7) d (cid:88) h = a h { v t h j , i , v t h i , j } j (cid:44) i = , It is unique in the quotient space S / D [ n ] (cid:39) C n − k OMBINATORICS OF DISCRIMINANTAL ARRANGEMENTS 11 then a = . . . = a d = . The following remark is a key point to prove the connection between linearly independence of K T -configurations andlinearly independence of associated K T -vector sets. Remark 4.3.
Let K T ( A t ) be the K T -configuration of the arrangement A t translated of A . Then for any c ∈ C , the K T -configuration K T ( A ct ) is an ”expansion” by c of K T ( A t ) , that is v cti , j = cv ti , j ( Figure 7 shows an example of expansionin the case of | T | = ). This is consequence of the fact that for any i ∈ [ r ] the vector OP cti joining the origin with thepoints P cti satisfies OP cti = cOP ti by definition of translation. Hence P cti P ctj = cP ti P tj , i.e.v cti , j = cv ti , j . Analogously we have that, if t , t ∈ C n are two translations thenv t i , j + v t i , j = v t + t i , j . P t P t P t P t K T ( A t ) v t , v t , v t , v t , v t , v t , P t P t P t P t K T ( A t ) v t , = cv t , v t , = cv t , v t , = cv t , v t , = cv t , v t , = cv t , v t , = cv t , F igure
7. All vectors v t i , j are obtained by multiplication of v t i , j by c , i.e. t = ct .We can now prove the main lemma of this section. Lemma 4.4.
Let A be a central generic arrangement of n hyperplanes in C k and T = { L , . . . , L r } be an r-set suchthat [ n ] = (cid:83) ri = L i . The K T -translated arrangements A t , . . . , A t d of A are linearly independent if and only if theirassociated K T -vector sets {{ v t h j , i , v t h i , j } j (cid:44) i } h = ,..., d are linearly independent.Proof. By definition A t , . . . , A t d are linearly independent if and only if translations t , . . . , t d are linearly independentvectors in C n / C . Let’s consider a linear combination (cid:80) dh = a h t h of vectors t h and translated arrangements A a h t h . ByRemark 4.3 we have that K T -vector sets associated to A a h t h verify d (cid:88) h = a h { v t h j , i , v t h i , j } j (cid:44) i = { v (cid:80) dh = a h t h j , i , v (cid:80) dh = a h t h i , j } j (cid:44) i . Since v ti , j is, by definition, the vector such that P ti + v ti , j = P tj then v ti , j = t is a translation such thatpoints P ti ≡ P tj coincides. Hence the condition that (cid:80) dh = a h { v t h j , i , v t h i , j } j (cid:44) i = (cid:80) dh = a h t h ∈ C . Indeed (cid:80) dh = a h { v t h j , i , v t h i , j } j (cid:44) i = { v (cid:80) dh = a h t h j , i , v (cid:80) dh = a h t h i , j } j (cid:44) i = (cid:80) dh = a h t h is a translation such that allintersection points P j coincide with the same point P i , i.e. P i = (cid:84) p ∈ (cid:83) ri = L i H (cid:80) dh = a h t h p is the center of the translatedarrangement A (cid:80) dh = a h t h of [ n ] = (cid:83) ri = L i hyperplanes. The proof of the statement follows. (cid:3) Let us remark that the assumption that the r -set T = { L , . . . , L r } satisfies the condition (cid:83) ri = L i = [ n ] in Lemma 4.4 isequivalent to consider, in the more general case in which (cid:83) ri = L i ⊂ [ n ], a subset A (cid:48) ⊂ A which only contains thehyperplanes indexed in (cid:83) ri = L i . On the other hand if a (central) generic arrangement A contains a subarrangement A (cid:48) which is non very generic then A is non very generic (this simply comes from the fact that non very genericity isa local property on the intersection lattice of the Discriminatal arrangement). Analogously, if there exists a restrictionarrangement A Y A(cid:48) = { H ∩ Y A (cid:48) | H ∈ A \ A (cid:48) } , Y A (cid:48) = (cid:84) H ∈A (cid:48) H of A which is non very generic, then A is non verygeneric and the following main theorem of this Section follows. Theorem 4.5.
Let A be a central generic arrangement of n hyperplanes in C k . If there exists a set T = { L , . . . , L r } with | (cid:83) ri = L i | = m and rank (cid:84) p ∈ (cid:84) ri = L i H p = y, which admits m − y − k − r (cid:48) independent K T -vector sets for some r (cid:48) < r,then A is non very generic.Proof. Let’s consider the subarrangement A (cid:48) of A given by hyperplanes indexed in the (cid:83) ri = L i and its essentialization,i.e. the restriction arrangement A (cid:48) Y , Y = (cid:84) p ∈ (cid:84) ri = L i H p . If y = rank Y then the arrangement A (cid:48) Y is a central essentialarrangement in C m − y , m = | (cid:83) ri = L i | . By Lemma 4.4, if A (cid:48) t , . . . , A (cid:48) t m − y − k − r (cid:48) are K T -translated of A (cid:48) Y associated to the m − y − k − r (cid:48) independent K T -vector sets, then A t , . . . , A t m − y − k − r (cid:48) are linearly independent vectors in S [ A (cid:48) Y ] / D [ m ] (cid:39) C m − y − k .That is A t , . . . , A t m − y − k − r (cid:48) span a subspace of dimension m − y − k − r (cid:48) . On the other hand, by construction, A t j are K T -translated, i.e. A t j ∈ ess ( X ) , X = (cid:84) ri = D L i for any j = , . . . , m − y − k − r (cid:48) , that is the space spanned by A t , . . . , A t m − y − k − r (cid:48) is included in ess ( X ). This implies that the simple intersection ess ( X ) has dimension d ≥ m − y − k − r (cid:48) > m − y − k − r that is its codimension is smaller than r , i.e. rank ess ( X ) < r and hence rank X < r . This implies that A (cid:48) is non verygeneric and hence A is non very generic. (cid:3) Theorem 4.5 allows to build non very generic arrangements simply imposing linear dependency conditions on vectors v i , j ∈ H i , j and, viceversa, to check wether an arrangement is non very generic by checking opportunely defined lineardependencies.Still two main questions are left open. One from geometric point of view and the other one combinatorial.(1) While we provided a geometric / algebraic necessary and su ffi cient condition for a simple intersection X ofmultiplicity r to be of rank X < r , it is still open the problem on non simple intersections. That is, is itpossible to have intersections (cid:84) mi = D S i , | S i | > k such that (cid:84) i ∈ I D S i (cid:44) D K for any subset I ⊂ [ m ] , | I |≥ (cid:84) mi = D S i < (cid:80) mi ( | S i | − k )? More precisely, does any such intersection contain a simple intersection ofrank strictly lesser than its multiplicity?(2) Which are the numerical conditions on the sets L (cid:48) i s for an intersection X to be simple and non very generic?In the next section we will provide non trivial examples of how to build non very generic arrangements by means ofTheorem 4.5. 5. E xamples of non very generic arrangements In this section we present few examples to illustrate how to use Theorem 4.5 to construct non very generic arrange-ments. To construct the numerical examples we used the software CoCoA-5.2.4 ( see [1]).
Example 5.1 ( B (12 , , A ) with an intersection of multiplicity 4 in rank 3) . Let L = [12] \ { , , } , L = [12] \{ , , } , L = [12] \ { , , } and L = [12] \ { , , } be subsets of [12] of k + = indices. It is an easy computationthat the set T = { L , L , L , L } is a -set. Let’s consider a central generic arrangement A of hyperplanes in C and A t an almost K T -translated, i.e. K T but for the hyperplane H and S = { L , L , L } . In this case m = n = , y = and m − k − r = − − = , hence, by Theorem 4.5 in order for A to be non very generic it is enoughthe existence of just one K T -vector set { v , , v , , v , } such that the vectors v , = v , − v , ∈ (cid:84) p ∈ L ∩ L \{ } H p and OMBINATORICS OF DISCRIMINANTAL ARRANGEMENTS 13 v , = v , − v , ∈ (cid:84) p ∈ L ∩ L \{ } H p belong to H (see Figure 8). Notice that since v , = v , − v , ∈ (cid:84) p ∈ L ∩ L \{ } H p ,v , ∈ H if v , , v , ∈ H . That is all hyperplanes in A can be chosen freely , but H which has to contain vectorsv , , v , . P P P P H t , H t , H t , (cid:84) p ∈ L ∩ L \{ } H tp (cid:84) p ∈ L ∩ L \{ } H tp (cid:84) p ∈ L ∩ L \{ } H tp v , v , v , v , v , v , F igure K T -configuration K T ( A t ) of B (12 , , A ) . v i , j are vectors in H i , j . Let’s see a numerical example. Let us consider hyperplanes of equation H i : α i · x = , with α i , i = , . . . , assignedas following: α = (0 , , , , , , − , , α = (0 , , , , , , , − , α = (0 , , − , , , , , ,α = (0 , , , , , , , , α = (0 , , , − , − , , , − , α = (0 , − , , , , − , − , ,α = (1 , , , , , − , − , , α = ( − , , , , , , , , α = ( − , , , , , − , , ,α = (1 , , , − , − , − , − , , α = (1 , , , , , , , . (8) In this case, we have the K T -vector set { v , , v , , v , } = { (1 , , , , , , , , (0 , , , , , , , , (0 , , − , , , , , } . The other vectors are obtained by means of relations v , = v , − v , , v , = − ( v , + v , ) , v , = − ( v , + v , ) , that is (9) v , = ( − , , , , , , , , v , = ( − , , , , , , , , v , = (0 , − , , , , , , and, finally, we get α = ( − , − , − , , , − , , by imposing the condition that α has to be orthogonal to v , andv , . Example 5.2 ( B (16 , , A ) with an intersection of multiplicity 4 in rank 3) . Let L = [16] \ { , , , } , L = [16] \ { , , , } , L = [16] \ { , , , } and L = [16] \ { , , , } be subsets of [16] of k + = indices. Theset T = { L , L , L , L } is a -set. Let’s consider a central generic arrangement A of 16 hyperplanes in C and A t be an almost K T -translated but for the hyperplane H and S = { L , L , L } . In this case m = n = , y = and m − k − r = − − = , hence, by Theorem 4.5 in order for A to be non very generic we need twolinearly independent K T -vector sets { v , , v , , v , } and { v , , v , , v , } such that the vectors v k , ∈ (cid:84) p ∈ L ∩ L \{ } H p andv k , ∈ (cid:84) p ∈ L ∩ L \{ } H p , k = , , belong to H (see Figure 9). Notice that since v k , = v k , − v k , ∈ (cid:84) p ∈ L ∩ L \{ } H p ,v k , ∈ H if v k , , v k , ∈ H . That is all hyperplanes in A can be chosen freely, but H which has to contain vectorsv k , , v k , , k = , . Here and in the rest of this section, freely means that we only impose the condition that A is a central generic arrangement. In particular thiscondition is always taken as given and imposed even if not written. P P P P H t , H t , H t , (cid:84) p ∈ L ∩ L \{ } H tp (cid:84) p ∈ L ∩ L \{ } H tp (cid:84) p ∈ L ∩ L \{ } H tp v k , v k , v k , v k , v k , v k , F igure K T -configuration K T ( A t ) of B (16 , , A ) . v ki , j are vectors in H i , j . Let’s see a numerical example. Let us consider hyperplanes of equation H i : α i · x = , with α i , i = , . . . , assignedas following. α = (0 , , , , , , , , , , − , α = (0 , , − , , , , , , , − , , α = (0 , , , , , , , , , , ,α = (0 , , , , , , , , , , , α = (0 , − , , , , , , , , − , , α = (0 , , , , , , , − , − , , ,α = (0 , , , , − , , − , , , , , α = (0 , − , , , , , , , , , , α = (1 , , , − , , , − , − , , , ,α = (2 , , , , , , , − , − , , , α = (3 , , , , , , , − , , , , α = (1 , , , , , , , , , , ,α = (1 , , , − , − , − , − , − , , − , − , α = (1 , , , , , , − , , − , , , α = (0 , , , − , − , − , , , − , − , . (10) In this case, we have the K T -vector sets { v , , v , , v , } = { (1 , , , , , , , , , , , (0 , , , , , , , , , , , (0 , , − , , , , , , , , } , { v , , v , , v , } = { (0 , , , , , , , , , , , (0 , , , , , , , , , , , (0 , , , , , − , , , , , } . The other vectors are obtained by means of relations v k , = v k , − v k , , v k , = − ( v k , + v k , ) , v k , = − ( v k , + v k , ) , k = , ,that is v , = ( − , , , , , , , , , , , v , = ( − , , , , , , , , , , , v , = (0 , − , , , , , , , , , , v , = (0 , , , − , , , , , , , , v , = (0 , , , − , , , , , , , , v , = (0 , , , , − , , , , , , and, finally, we get α = (1 , , , − , − , − , , , , , by imposing the conditions that α has to be orthogonal tov k , and v k , , k = , . Example 5.3 ( B (10 , , A ) with an intersection of multiplicity 5 in rank 4) . Let L = { , , , } , L = { , , , } , L = { , , , } , L = { , , , } and L = { , , , } be subsets of [10] of k + = indices. The set T = { L , L , L , L , L } is a -set. Let’s consider a central generic arrangement A of 10 hyperplanes in C and A t be almost K T -translatedbut for the hyperplane H and S = { L , L } . In this case m = n = , y = and m − k − r = − − = ,hence, by Theorem 4.5 in order for A to be non very generic we need three linearly independent K T -vector sets { v , , v , , v , , v , } , { v , , v , , v , , v , } and { v , , v , , v , , v , } such that the vectors v k , , k = , , , belong to H (seeFigure 10). Notice that since in this case hyperplanes are planes, then the three vectors v ki , j , k = , , have to belinearly dependent for any choice of indices ( i , j ) , i (cid:44) j. This additional condition forces that at most 8 hyperplanesin A can be chosen freely, while both H and H have to contain the dependent vectors v k , and v k , , k = , , ,respectively. OMBINATORICS OF DISCRIMINANTAL ARRANGEMENTS 15 P P P P P H t , H t , H t , H t , H t , H t , H t , H t , v k , v k , v k , v k , F igure K T -configuration K T ( A t ) of B (10 , , A ) . v ki , j is a vector in H i , j . Let’s see a numerical example. Let us consider hyperplanes of equation H i : α i · x = , with α i , i = , . . . , assignedas following. α = (0 , , , α = (20 , , − , α = (2 , − , ,α = (3 , , , α = (0 , , , α = (1 , − , ,α = (1 , , , α = (4 , − , − . (12) In this case, we have the K T -vector sets { v , , v , , v , , v , } = { (1 , − , , ( 92 , , , ( 92 , ,
252 ) , ( − , , − } , { v , , v , , v , , v , } = { ( − , , − , ( − , − , − , ( − , − , − , ( − , , −
503 ) } , { v , , v , , v , , v , } = { ( − , , − , ( − , − , − , ( − , − , − , ( − , , −
716 ) } . The other vectors are obtained by means of relations v ki , l = v k , l − v k , i , where ≤ i < l ≤ , k = , , , that isv , = ( 72 , , , v , = ( 72 , ,
52 ) , v , = ( − , , − , v , = (0 , − ,
52 ) , v , = ( − , , − , v , = ( − , , − , v , = ( − , − , , v , = ( − , − , − , v , = ( 43 , − ,
103 ) , v , = (6 , , − , v , = ( 253 , ,
103 ) , v , = ( 73 , ,
313 ) , v , = ( − , − , , v , = ( 16531040 , − , − , v , = ( 53 , , −
116 ) , v , = ( 32131040 , , − , v , = ( 196 , , −
116 ) , v , = ( 2413120 , , . (13) Finally, we get α = (314 , − , − and α = (139 , , − by imposing the conditions that α and α have to beorthogonal to v k , and v k , , k = , , . Remark 5.4.
Notice that Example 5.3 is slightly di ff erent from other examples for two reasons. Firstly, it uses adi ff erent combinatorics. In the Examples 5.1 and 5.2 the 4-sets T = { L , L , L , L } are of the form L i = [ n ] \ K i with K i ’s which satisfy the properties (cid:83) i = K i = [ n ] and K i ∩ K j = ∅ while in the Example 5.3 they are not. Secondly, in theExamples 5.1 and 5.2 in order to obtain non very generic arrangement we could choose all hyperplanes freely but one,while in the Example 5.3 two hyperplanes had to be fixed as a result of the need of three independent K T -vector sets intwo dimensional hyperplanes. Indeed this dependency condition gives rise to independent equations of the form (14) v i , j = α v i , j + β v i , j which fix the entries of the vectors v k , i , i = , , uniquely for any choice of three dependent vectors v k , , k = , , .Hence the vectors v k , and v k , , k = , , are determined and so are the two hyperplanes H and H . Remark 5.5.
Notice that the above examples are just very special cases of r-sets T . How to describe the r-sets T thatcan give rise to ( simple ) non very generic intersections is an open problem. R eferences [1] J. Abbott and A. M. Bigatti, CoCoA: a system for doing Computations in Commutative Algebra, available at http: // cocoa.dima.unige.it.[2] C. A. Athanasiadis, The Largest Intersection Lattice of a Discriminantal Arrangement, Contributions to Algebra and Geometry, Vol. 40, No. 2(1999).[3] M. Bayer, K. Brandt, Discriminantal arrangements, fiber polytopes and formality, J. Algebraic Comb., 6 (1997), pp. 229-246.[4] H. Crapo, The combinatorial theory of structures, in: “Matroid theory” (A. Recski and L. Loca´sz eds.), Colloq. Math. Soc. J´anos Bolyai Vol.40, pp. 107-213, North-Hokkand, Amsterdam-New York, 1985.[5] H. Crapo and G.C. Rota, The resolving bracket. In Invariant methods in discrete and computational geometry (N. White ed.), pp. 197-222,Kluwer Academic Publisher, Dordrecht 1995.[6] M. Falk, A note on discriminantal arrangements, Proc. Amer. Math. Soc., 122 (4) (1994), pp. 1221-1227.[7] S. Felsner and G.M. Ziegler, Zonotopes associated wuth hugher Bruhat orders, Discrete Mathematics, 241 (2001), pp. 301-312 .[8] M. Kapranov, V. Voevodsky, Free n-category generated by a cube, oriented matroids, and higher Bruhat orders Funct. Anal. Appl., 2 (1991), pp.50-52.[9] M. Kapranov, V. Voevodsky, Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results).Cahiers de Topologie et Geometrie dijferentielle categoriques, 32 (1991), pp. 11-27.[10] M. Kapranov, V. Voevodsky, Braided monoidal 2-categories and Manin–Schechtman higher braid groups, J. Pure Appl. Algebra, 92 (3) (1994),pp. 241-267.[11] T.Kohno, Integrable connections related to Manin and Schechtman’s higher braid groups, Illinois J. Math. 34, no. 2, (1990) pp. 476–484.[12] R.J. Lawrence, A presentation for Manin and Schechtman’s higher braid groups, MSRI pre-print(http: // / ∼ ruthel / papers / premsh.html) (1991).[13] A. Libgober and S. Settepanella, Strata of discriminantal arrangements, Journal of Singularities Volume in honor of E. Brieskorn 18 (2018).[14] Yu. I. Manin and V. V. Schechtman, Arrangements of Hyperplanes, Higher Braid Groups and Higher Bruhat Orders, Advanced Studies in PureMathematics 17, (1989) Algebraic Number Theory in honor K. Iwasawa pp. 289-308.[15] P. Orlik, Introduction to Arrangements, CBMS Reg. Conf. Ser. Math., vol. 72, American Mathematical Society, Providence, RI, USA (1989).[16] P. Orlik, H. Terao, Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences], vol. 300, Springer-Verlag, Berlin (1992).[17] M. Perling, Divisorial cohomology vanishing on toric varieties, Doc. Math., 16 (2011), pp. 209-251.[18] S. Sawada, S. Settepanella and S. Yamagata, Discriminantal arrangement, 3 × Gr (3 , C n ), Ars Mathematica Contemporanea, Vol 16, No 1,(2019), pp. 257-276.[20] G.M. Ziegler, Higher Bruhat orders and cyclic hyperplane arrangements, Topology 32, (1993), pp. 259–279.D epartment of M athematics , H okkaido U niversity , J apan . Email address : [email protected] Email address : [email protected]