Table Of ContentSTRATA OF DISCRIMINANTAL ARRANGEMENTS
ANATOLYLIBGOBERANDSIMONASETTEPANELLA
6
1 Abstract. We give explicit description of the multiplicities of codimension
0 two strata of discriminantal arrangements of Manin and Schechtman. As
2 applications we discuss connection of these results with properties of Gale
transform, the study of the fundamental groups of the complements to dis-
n criminantal arrangements and study of the space of generic arrangements of
a linesinprojectiveplane.
J
5
2
1. Introduction
]
O
In 1989 Manin and Schechtman (cf.[15]) considered a family of arrangements
C
of hyperplanes generalizing classical braid arrangements which they called the dis-
h. criminantal arrangements (cf. [15] p.209). Such an arrangement B(n,k),n,k ∈ N
t for k ≥2 depends on a choice H0,...,H0 of collection of hyperplanes in general po-
a 1 n
m sition in Ck. It consists of parallel translates of H1t1,...,Hntn,(t1,...,tn)∈Cn which
fail to form a generic arrangement in Ck. B(n,k) can be viewed as a generalization
[
ofthepurebraidgrouparrangements(cf. [18])withwhichB(n,1)coincides. These
1 arrangementshaveseveralbeautifulrelationswithdiverseproblemsincludingcom-
v binatorics (cf. [15] and also [5], which is an earlier appearance of discriminantal
5
arrangmements), Zamolodchikov equation with its relation to higher category the-
7
ory (cf. Kapranov-Voevodsky [10]), and vanishing of cohomology of bundles on
4
6 toric varieties (cf. [19]).
0 ThepurposeofthisnoteistostudydependenceofB(n,k)onthedataH0,...,H0.
1 n
1. Paper [15] concerns with arrangements B(n,k) which combinatorics is constant on
0 a Zariski open set Z in the space of generic arrangements H0,i=1,...,n but does
i
6 not describe the set Z explicitly. It was shown in [7] 1 that, contrary to what was
1
frequentlystated(seeforinstance[17], sect. 8, [18]or[12]), thecombinatorialtype
:
v ofB(n,k)indeeddependsonthearrangementA0 ofhyperplanesH0,i=1,...,nby
i
i providinganexampleofA0forwhichthecorrespondingdiscriminantalarrangement
X
has combinatorial type distinct from the one which occurs when A0 varies within
r
a the Zariski open set Z. In [1] Athanasiadis gives an algebraic condition for A0
to yield the discriminantal arrangement B(n,k) with combinatorics which occurs
when A0 belongs to Z.
Our main result describes a simple necessary and sufficient geometric condition
on arrangement A0 which assures that multiplicities of strata of corresponding
B(n,k) up to codimension 2 are generic i.e. the same as occur for A0 ∈ Z. This
result casts a light on the geometric reasons behind the algebraic condition in [1]
1991 Mathematics Subject Classification. 52C3552B3520F3614-XX05B35.
Keywordsandphrases. discriminantalarrangements,braidgroups,fundamentalgroups,Gale
transform.
ThefirstnamedauthorwassupportedbyagrantfromSimonsFoundation.
1thisisalsocontentofearlierwork[5]sect. 4
1
2 ANATOLYLIBGOBERANDSIMONASETTEPANELLA
and holds in case of arrangements defined in any field, including complex one.
Moreoveritissuggestiveofpossiblegeneralisationstohighercodimensionalstrata.
Wegiveseveralapplicationsofourresult. Firstly,weinterpretitintermsofGale
transform (cf. section 4). Relation between discriminantal arrangements and Gale
tranformcanbeseen,atleastimplicitly,alreadyinpaper[7]. Fromthisviewpoint
ourresultcanbeviewedasinvarianceofcertaintypeofcollinearity: specialposition
of data of A0 is equivalent of presence of dependencies in the Gail transform which
in turn is equivalent to presence of strata of multiplicity 3 in arrangement B(n,k).
We shall give independent verification of such invariance using interpretation of
Gale transform of six-tuple of point in P2 using cubic surfaces (cf. [6]): the strata
of B(6,3) having multiplicity 3 correspond cubic surfaces admitting an Eckardt
point (cf. section 4.2).
Other applications have to do with the fundamental groups of the complements
todiscriminantalarrangements. WesupplementR.Lawrencepresentation(cf. [12])
bygivingpresentationofthefundamentalgroupinthecaseofnon-genericdiscrim-
inantal arrangements (i.e. for which A0 ∈/ Z). In fact we give calculations yielding
the braid monodromy and presentation of discriminantal arrangement.
Finally we show that in the case of discriminantal arrangements of lines in the
plane, there is only one combinatorial type of discriminantal arrangements. We
alsoshowthatthisinturnsopensthedoortodescriptionofthefundamentalgroup
of the space of generic arrangements in P2, i.e. space of ordered collections of lines
which are in general position, in terms of the braid groups and the fundamental
groupsof discriminantalarrangement (cf. [9] foradiscussion ofanalyticproperties
of spaces of generic arrangements)
The content of paper is the following. In section 2 we introduce several notions
usedlaterandrecalldefinitionsfrom[15]. Section3containsoneofthemainresults
ofthispaperdescribingcodimension2strataofdiscriminantalarrangementshaving
multiplicity 3 and showing absence of codimension 2 strata having multiplicity 4
(withobviousexceptions,cf. Theorem3.8). Thesection4includesanassortmentof
corollaries: itscontentisapparentfromtheheadingsofeach. Thelasttwosections
describe independence of combinatorics of the B(n,2) from initial arrangement A0
and the fundamental groups of the complements to discriminantal arrangements
and the space of generic arrangements of lines in P2.
Finally, the first named author wants to that I.Dolgachev for a useful comment
onmeterialinsection4authorswanttothankMaxPlanckInstituteandUniversity
ofHokkaido2forhospitalityduringvisitstothisinstitutionswheremuchofthework
on this project was done
2. Preliminaries
2.1. Discriminantal arrangements. Let H0,i = 1,...,n be a generic arrange-
i
mentinCk,k <ni.e. acollectionofhyperplanessuchthatdim(cid:84) H0 =
i K,CardK=k i
0. SpaceofparalleltranslatesS(H0,...,H0)(orsimplySwhendep∈endenceonH0 is
1 n i
clearornotessential)isthespaceofn-tuplesH ,...,H suchthateitherH ∩H0 =∅
1 n i i
or H =H0 for any i=1,...,n. One can identify S with n-dimensional affine space
i i
Cn in such a way that (H0,...,H0) corresponds to the origin.
1 n
We will use the compactification of Ck viewing it as Pk \ H endowed with
collection of hyperplanes H¯0 which are projective closures of affi∞ne hyperplanes
i
2ThiswassupportedbyJSPSKakenhiGrantNumber26610001.
STRATA OF DISCRIMINANTAL ARRANGEMENTS 3
H0. Condition of genericity is equivalent to (cid:83) H0 being a normal crossing divisor
i i i
in Pk. The space S can be identified with product L ×...×L where L (cid:39) C is
1 n i
parametrizedbyP1 pencilofhyperplanesspannedbyH andH0 withthedeleted
i
point corresponding to H and the origin correspondin∞g to H0. In particular, an
i
ordering of hyperplanes in∞A determines the coordinate system in S.
For a generic arrangement A in Ck formed by hyperplanes H ,i = 1,...,n the
i
traceatinfinity(denotedbyA )isthearrangementformedbyhyperplanesH =
,i
H¯0 ∩ H . For an intersecti∞on H ∩ ... ∩ H ,t ≥ 1 of hyperplanes in A∞, we
cain cons∞ider the arrangement in thi1is intersectiiton formed by intersections of the
hyperplanes H ∈A,H (cid:54)=H ,s=1,...,t with H ∩...∩H . We call the resulting
is i1 it
arrangement the restriction of A to H ∩...∩H .
i1 it
The trace A of an arrangement A determines the space of parallel translates
S(H0,...,H0) (a∞s a subspace in the space of n-tuples of hyperplanes in Pk).
1 n
For generic arrangement A , we consider the closed subset of S(H0,...,H0)
1 n
formed by those collections wh∞ich fail to form a generic arrangement. This sub-
set is a union of hyperplanes with each hyperplane corresponding to a subset
K = [i ,...,i ] ⊂ [1,...,n] and consists of n-tuples of translates of hyperplanes
1 k+1
H0,...,H0 in which translates of H0,...,H0 fail to form a generic arrangement.
1 n i1 ik+1
(equations are given by (3) below). It will be denoted D . The corresponding ar-
K
rangement will be denoted B(n,k,A ) and called the discriminantal arrangement
corresponding to A . ∞
The cardinality∞of B(n,k) is equal to (cid:0) n (cid:1). Each hyperplane D contains
k+1 K
the k-dimensional subspace T of S(H0,...,H0) formed by n-tuples of hyperplanes
1 n
containing a fixed point in Ck. Clearly, the essential rank (cf. [21]) of B(n,k,A )
is n−k and the arrangement induced by the arrangment of hyperplanes D ∞in
K
the quotient of S(H0,...,H0) by T is essential. It is called the essential part of the
1 n
discriminantal arrangement.
2.2. HyperplanesinB(n,k). RecallthatarbitraryarrangementAofhyperplanes
W ,...,W ⊂ Ck defines canonical stratification of Ck in which strata are defined
1 N
as follows. Let L(A) be the intersection poset of subspaces in Ck each being the
intersection of a collection of hyperplanes chosen among W ,...,W and for each
1 N
P ∈ L(A), let Σ ⊂ [1,...,N] be the set of indices of hyperplanes W such that
P i
P = ∩ W . Vice versa, given a subset Σ ⊂ [1,...,N] we denote by w the
subspacie∈ΣwP =i ∩ W . Stratum of P is the submanifold of Ck defined as foΣllows:
Σ i Σ i
∈
(cid:91)
(1) S =P \ w .
P Σ
ΣP⊂Σ
If an arrangement A = {W ,...,W } in Ck is generic then the finite subset in
1 N
Ck consisting of 0-dimensional strata has cardinality (cid:0)N(cid:1) and its elements are in
k
one to one correspondence with the subsets of [1,...,N] having cardinality k.
The multiplicity of points p ∈ S considered as a point on the subvariety
P
(cid:83) W in Ck is constant along the stratum. We call it the multiplicity of
i=1,...,N i
the stratum S . It is equal to cardinality of the set Σ .
P P
Aswenoted,thehyperplanesofB(n,k)correspondtosubsetsofcardinalityk+1
in [1,...,n]. Their equations can be obtained as follows. Let K,|K|=k+1, be a
subset in [1,...,n] and let
(2) αjy +...+αjy =x0, j ∈[1,...,n]
1 1 k k j
4 ANATOLYLIBGOBERANDSIMONASETTEPANELLA
beequationofhyperplaneH0ofarrangementA0 ={H0,...,H0}∈Cn\B(n,k,A )
j 1 n ∞
in selected coordinates y ,...,y in Ck. The hyperplanes H , j ∈ K, of arrange-
1 k j
ment in S with equations αjy +...+αjy = x , j ∈ K, will have non empty
1 1 k k j
intersection iff
α1 ... α1 x
1 k 1
(3) det ... ... ... ... =0
αk+1 ... αk+1 x
1 k k+1
This provides a linear equation in x ,j = 1,...,k +1, for hyperplane D corre-
j K
sponding to K.
Let J be a subset in {1,...,n} of cardinality a,
(4) D ={(H ,...,H )∈S such that ∩ H (cid:54)=∅}
J 1 n i J i
∈
and
(5) P (J)={K ⊂J such that |K |=k+1} .
k+1
Then
(cid:92)
(6) D = D
J K
K∈Pk+1(J)
is intersection of (cid:0) a (cid:1) hyperplanes. In particular D , |J|≥k+1, is a linear sub-
k+1 J
space and the multiplicity of (cid:83) D at its generic point is (cid:0) a (cid:1). Moreover,
K=k+1 K l+1
codim D is a−k. | |
J
2.3. Projections of discriminantal arrangements. . Let Ξ ⊂ [1,...,n] be a
subset of the set of indices and let S(Ξ) ⊂ S be subspace of the space of trans-
lates of hyperplanes of a generic arrangement H0,...,H0 consisting of translates
1 n
of hyperplanes with indices in Ξ. One has projection p : S → S(Ξ) obtained by
Ξ
omittingfromacollectionsoftranslated,thetranslatesofhyperplaneswithindices
outside of Ξ. The image of a subspace D ,J ⊂ [1,...,n] is a proper subspace iff
J
CardJ ∩Ξ≥k+1 and in fact p (D )=D . Moreover, p is a stratified map,
Ξ J J Ξ Ξ
with both S and S(Ξ) endowed with stratifica∩tions corresponding to discriminantal
arrangements, in the sense that image of a stratum in S belongs to a stratum in
S(Ξ).
The maps p are locally trivial fibrations if and only if k = 1. Note, that some
Ξ
recent works (cf. for example [8]) refer to discriminantal arrangements in a more
narrowsensethanusedinthispaperi.e. asthearrangementsinducedonthefibers
of p given explicitly as
Ξ
(7)
p−{11,...,l}(t1,...,tl)={(z1,...,zn−l)|zi =zj or zi =tk, k =1,...,l, i,j =1,...,n−l}
3. Codimension two strata having multiplicity 3
In this section we shall describe necessary and sufficient conditions on the trace
atinfinityA sothatthecorrespondingdiscriminantalarrangementwillhavecodi-
menion two∞strata having multiplicity 3. We shall start with two lemmas (3.1 and
3.4), providingexamplesofnon-transversaltriplesofhyperplanesD (subscriptof
K
a hyperplane always is a set of cardinality k+1).
STRATA OF DISCRIMINANTAL ARRANGEMENTS 5
Lemma 3.1. Let s ≥ 2, n = 3s,k = 2s − 1 and K ,i = 1,2,3, be subsets of
i
(cid:84)
[1,...,n] such that Card K =2s,Card K ∩K =s, i(cid:54)=j, K =∅ (in particular
i i j i
(cid:83)
Card K =n=3s).
i
Let A0 be a generic arrangement of n hyperplanes in Ck. Consider the triple of
codimension s subspaces H0 = H0 ∩...∩H0 ∩H , i ,...,i ∈ K ∩K ,i (cid:54)= j
of the hyperplane at infinity∞H,i,j i.e.i1the closuriess of i∞rredu1ciblescompoinentsjof the
codimension s strata of the trac∞e of A0 at infinity.
For each K , i = 1,2,3, let D ⊂ Cn denote the hyperplane in B(3s,2s−1,A0)
i Ki
corresponding to the subset of indices K .
i
If subspaces H0 ⊂H span a proper subspace in H then Codim (cid:84)D =2.
Otherwise this∞co,id,jimens∞ion is equal to 3. ∞ Ki
This lemma suggests the following:
Definition 3.2. A generic arrangement in P2s 2,s ≥ 2, is called dependent if
−
it is composed of 3s hyperplanes W which can be partitioned into 3 groups, each
i
containing s hyperlanes, such that 3 subspaces of dimension s − 2 , each being
intersection of hyperplanes in one group, span a proper subspace in P2s 2 3. We
−
call these three s−2-dimensional subspaces dependent.
Proof of Lemma 3.1. Consider first the case when subspaces H0 span a proper
,i,j
hyperplaneinH whichweshalldenoteH. NotethatdimH =∞2s−2,dimH0 =
,i,j
s−2 and as a co∞nsequence of A being generic, these subspac∞es do not intersec∞t i.e.
the subspace which they span has dimension greater than 2s−4.
Let A = {H0} be arrangement in Ck = C2s 1 which belongs to D and
D . Hence (cid:84)i H0 (cid:54)= ∅ and (cid:84) H0 (cid:54)= ∅−. We show that (cid:84) KH1 0 (cid:54)=
∅ Kw2hich would iim∈Kp1ly ithat Codim(cid:84)i∈K2 Di = Codim(cid:84) D i=∈K32. iLet
i=1,2,3 Ki i=1,2 Ki
L = (cid:84) H0. Each L is a codimension s subspace in C2s 1 and
Li,j ∩H s∈=KHi∩0Kj,i<.jSinsce A∈D i,j, the subspaces L and L have non−empty
init,ejrsect∞ion. Th∞e,rie,jforetheyspanaKh1yperplaneL inC12,2s 1 whic1h,3intersectionwith
1 −
H is the hyperplane H spanned by H and H . The hyperplane L is
,1,2 ,1,3 1
sp∞anned by the intersection point L ∩∞L and the∞hyperplane H.
1,2 1,3
Similarly, since A∈D both L and L have a point in common they span
K2 1,2 2,3
the hyperplane L spanned by this point and the above hyperplane H which can
2
be described as the plane spanned by H and H . Both hyperplane L and
,1,2 ,1,3 1
L contain L and hence coincide. He∞nce L an∞d L being both in L = L
2 1,2 1,3 2,3 1 2
(cid:84)
must have a point in common and hence A∈ D .
i=1,2,3 Ki
Now assume that H span H . Let A ∈ D ∩D be sufficiently generic
in this space. We sho∞w,it,jhat it do∞es not belongKt1o D K.2 Consider family of s
K3
codimensionalsubspacesinC2s 1 whichcompactificationintersectsthehyperplane
−
atinfinityatH0 . SelectionofAdeterminessubspacesL ,L ⊂C2s 1 which
,2,3 1,2 1,3 −
have a common∞point and moreover the subspace L which intersects L . Since
2,3 1,2
tripleH isnotinahyperplaneinH ,soisthecaseforsubspaceL ,L and
,i,j 1,2 1,3
H i∞n the compactification of C2s 1∞. Hence generic subspace L of codimension
,2,3 −
s ∞containing H and intersecting L will have empty intersection with L .
,2,3 1,2 1,3
The correspond∞ing arrangement A having L as the subspace L will not belong
(cid:48) 2,3
to D but will be in D ∩D .
K3 K1 K2
3with this terminology, the assumption of lemma 3.1 is that the trace of A0 is dependent
genericarrangement
6 ANATOLYLIBGOBERANDSIMONASETTEPANELLA
H
∞
H∞,2,3 H∞,1,2 H40
H∞,1,3 L1,2
l
H30
H10
L1,3
H60
H20 L2,3
H50
Figure 1.
(cid:3)
If s=2 in Lemma 3.1 then H are points in the 2 dimensional space P2 and
,i,j
conditiontospanapropersubspa∞ceinH ,i.e. beingco-linearpoints,corresponds
tothecaseofFalk’sexampleofspecialdi∞scriminantalarrangementin[7]. Weshall
illustrate the argument in Lemma 3.1 in this particular case.
Example 3.3. Let consider the case n=6 and k =3, that is generic arrangement
A0 ={H0,...,H0} in C3. In Lemma 3.1 this corresponds to s=2 and, after pos-
1 6
sible relabelling, K =(1,2,3,4),K =(3,4,5,6),K =(1,2,5,6). Then subspaces
1 2 3
L = (cid:84) H0 are lines L = H0∩H0,L = H0∩H0,L = H0∩H0.
Ini,jthis cass∈eKii.∩eK.jwhsen dimH =1,32 the 1assert2ion1o,2f Lemm3 a 3.14 is2t,3hat the5 poin6ts
H =L ∩H span a li∞ne l in H i.e. the points H are collinear if and
,i,j i,j ,i,j
on∞ly if dim (cid:84)∞D =4 (see Figur∞e 1). ∞
i=1,2,3 Ki
AnarrangementA={Ht1,...,Ht6}oftranslatesofplanesfromA0 isapointin
1 6
D ∩D iffpairwiseintersectionsinC4 oflinesl =Ht1∩Ht2,l =Ht3∩Ht4
K1 K2 1,3 1 2 1,2 3 4
andl =Ht5∩Ht6 arenonempty. Weclaimthatcolinearityconditionimpliesthat
2,3 5 6
these three lines belong to the same plane. Indeed, since A consisits of translates
STRATA OF DISCRIMINANTAL ARRANGEMENTS 7
of planes in A0 line l has same point at infinity H as does the line L .
i,j ,i,j i,j
Condition that H span a line l ∈ H implies that∞any plane containing two
,i,j
lines l intersects∞H in l. That is two∞planes containing above pairs of lines l
i,j i,j
are coincident. This i∞mplies that lines l and l have non-empty intersection i.e.
1,2 2,3
(cid:84) H0 (cid:54)=∅ and hence A∈D .
i=1,2,5,6 i K3
Viceversa,ifpointsH aren’tcolinearthenitispossibletofindconfigurations
,i,j
in which, for example, l∞ intersects both l and l , but l ∩ l = ∅, i.e.
1,3 1,2 2,3 1,2 2,3
A∈D ∩D and A∈/ D .
K1 K3 K2
Via restriction of arrangements, the lemma 3.1 leads to other examples of dis-
criminantal arrangements having codimension two strata with multiplicity 3.
Lemma 3.4. Let A0 be a generic arrangement in Ck such that its trace at infinity
A containsthyperplanesH ,...,H forwhichtherestrictionofA onH ∩
,1 ,t ,1
...∞∩H is a dependent (cf∞. Def. 3∞.2) arrangement of 3s = n−t ∞hyperpla∞nes.
,t
Then B∞(n,k,A ) admits a codimension two stratum of multiplicity 3.
∞
Proof. Itfollowsfromlemma3.1thatA0containscollectionofhyperplanesH ,...,H
1 t
such that the discriminantal arrangement in H ∩...∩H corresponding to inter-
1 t
section of the trace of A with H ∩...∩H admits a codimension 2 stratum
,1 ,t
having multiplicity 3. The dimen∞sion of this∞stratum is 3s−2 where n−t = 3s
and k −t = 2s−1. Let K ,i = 1,2,3 be subsets of [t+1,...,n] with the prop-
i
erty as in lemma 3.1. In particular CardK = 2s = k−t+1. The above 3s−2
i
dimensional stratum of discriminantal arrangement of n−t hyperplanes in Ck t
−
is the transversal intersection of two submanifolds of Cn. One is the stratum of
discriminantal arrangement B(n,k,A ) having dimension 3s − 2 + t formed by
hyperplanes D ,i=1,2,3, and∞another is the intersection of t hyperplanes
in S(H0,...,H0K)i∪d[e1fi,..n.,etd] by vanishing of coordinates corresponding to H0,...,H0.
1 n 1 t
Hence the multiplicity of this stratum of B(n,k,A ) equals 3. This yields the
lemma. ∞ (cid:3)
Corollary 3.5. Ifk ≥3andn≥k+3thenthereexistsagenericarrangementofn
hyperplanes in Ck such that the corresponding discriminantal arrangement admits
a codimension two stratum of multiplicity 3.
Proof. Lemma3.4impliesthatforapair(n,k)suchthatexistintegerst≥0,s≥2
satisfying
(8) n=3s+t k =2s−1+t
thereexistsagenericarrangementA0ofnhyperplanesinCk suchthatrestrictionof
trace H ∩A0 of A0 on intersection of its t hyperplanes is dependent. By Lemma
3.4 the∞discriminantal arrangement corresponding to such A will admit required
stratum. Given (n,k)∈N2, the relation (8) has unique solution s=n−k−1,t=
3k−2n+3 which satisfies s≥2,t≥0 iff
3
(9) k+3≤n≤ (k+1), k ≥3
2
Given an arrangement B(n,k,A ) admitting codimension 2 strata of multiplicity
3, extension of A to arrangem∞ent of N ≥ n hyperplanes by adding sufficiently
generic hyperplan∞es yields an arrangement A such that B(N,k,A ) contains
(cid:48) (cid:48)
B(n,k,A ) as a subarrangement and hence a∞dmits strata of codime∞nsion 2 and
multiplici∞ty 3. On the other hand, for n = k +2, B(k +2,k,A ) has only one
∞
8 ANATOLYLIBGOBERANDSIMONASETTEPANELLA
stratum of multiplicity k+2 i.e. the inequality n (cid:54)= k+3 is sharp. Case k = 2
considered in section 5. (cid:3)
An example, illustrating above two lemmas as well as further cases, is the fol-
lowing.
Example 3.6. Let A = (cid:83)H ,i = 1,...,8 be generic arrangmenent in P4.
,i
A defines the arrange∞ment of li∞nes H ∩H ∩H ,i = 1,...,6 in the plane
,i ,7 ,8
H∞ ∩H . The induced arrangemen∞t A | ∞ ∞is generic since A is.
∞A,7ssume∞t,h8atthedoublepointsH ∩H ∞∩H∞H,7∩H∩∞,H8 , H ∩H ∩∞H ∩
,1 ,2 ,7 ,8 ,3 ,4 ,7
H , H ∩H ∩H ∩H ∞are co∞llinear.∞ ∞ ∞ ∞ ∞
,8 ,5 ,6 ,7 ,8
∞Consid∞er the h∞yperpla∞nes ∞
D ,D ,D
1,2,3,4,7,8 3,4,5,6,7,8 1,2,5,6,7,8
in discriminantal arrangement B(8,5) corresponding to such A and the hyper-
planes D ,D ,D in the discriminantal arrangemen∞t in 3-space H ∩
1(cid:48),2,3,4 3(cid:48),4,5,6 1(cid:48),2,5,6 7
H for a generic choice of hyperplanes H ,H intersecting the hyperplane at infin-
8 7 8
ity at H ,H respectively. Then arrangement of 6 hyperlanes in C5 including
,7 ,8
H ,H h∞as a c∞ommon point if and only if the arrangement of 4 planes in 3-space
7 8
H ∩H has a common point. Hence
7 8
(10)
dimD ∩D ∩D =2+dimD ∩D ∩D =6
1,2,3,4,7,8 3,4,5,6,7,8 1,2,5,6,7,8 1(cid:48),2,3,4 3(cid:48),4,5,6 1(cid:48),2,5,6
(the last equality uses the Example 3.3). Hence the discriminantal arrangement
with such arrangement A has a codimension two stratum of multiplicity 3.
This case illustrates the ∞case considered in Theorem 3.8 (2) t = 2 below, corre-
sponding to hyperplanes H ,H , the induced arrangement in P5 2 1 given by
,7 ,8 − −
hyperplanes H ∩H ∩∞H ∞, i=1,...,6 and s=2.
,i ,7 ,8
∞ ∞ ∞
The next Lemma will be useful to prove the absence of codimension 2 strata of
multiplicity 4.
Lemma 3.7. For s≥2, there is no quadruple of subspaces V ⊂P3s 2,i=1,2,3,4
i −
having dimension 2s−2 such that intersection of each V with each of remaining 3
i
consists of 3 subspaces P =V ∩V ,i(cid:54)=j of V and such that
i,j i j i
a) each P has dimension s−2
i,j
b) any pair P ,P ,i(cid:54)=j (cid:54)=k spans a hyperplane in V 4 and
i,j i,k i
c) all three, P ,P ,P belong to a hyperplane in V .
i,j i,k i,l i
Proof. Weshallstartwiththecases=2. Assumethataconfigurationasinlemma
does exist and consider a quadruple of planes V ,i=1,...,4 in P4 such that
i
a) any two intersect at a single point,
b) all 6 points P =V ∩V ,P =P obtained in this way are distinct, and
i,j i j i,j j,i
c) the intersection points of any plane V with remaining 3 are colinear.
i
For a fixed k, the triple of points P ,i (cid:54)= j (cid:54)= k outside of V , determines the
i,j k
triple of lines L ⊂V ,i(cid:54)=k spanned by points P ,P ,i(cid:54)=j (cid:54)=l (cid:54)=k. These lines
i i i,j i,l
L ,i (cid:54)= k by their definition are pairwise concurrent (L ∩L = P ) and hence
i i j i,j
belong to a plane H. By assumption c), the plane V intersects V ,i(cid:54)=k at a point
k i
on the line L distinct from P ,P ,i (cid:54)= j (cid:54)= k i.e. having a common point with
i i,j i,k
4dimension of the span of (s−2) dimensional subspaces Pi,j and Pi,k in 2s−2 dimensional
spaceVi doesnotexceed2s−3.
STRATA OF DISCRIMINANTAL ARRANGEMENTS 9
H on the line L . Hence H and V have 3 distinct non-colinear points in common
i k
and therefore H =V but this contradict to dim V ∩V =0.
k k i
Now consider the case s > 2. Similarly to above, s−2 dimensional subspaces
P =V ∩V of P3s 2 determine the subspaces L ⊂V ,i(cid:54)=k (for a fixed k) each
i,j i j − i i
being spanned by pairs P ,P ,i,j,l (cid:54)= k which are outside of V . Each L is a
i,j i,l k i
hyperplaneinV (i.e. dimL =2s−3). Moreover,thedimensionofthesubspaceH
i i
ofP3s 2spannedbyL ,L ,i,j,l(cid:54)=k,is3s−4. H canbedescribedasthesubspace
− i,j i,l
ofP3s 2 spannedbytripleofsubspacesP ,i,j (cid:54)=k. Nowbyourassumptionc),V
− i,j k
containsans−2dimensionalsubspaceofL ,i,j (cid:54)=ki.e. P . Thesubspacehence
i,j i,k
is also a subspace of H. This implies that V ⊂ H. The dimension of intersection
k
L and V which are both subspaces of H is (2s−3)+(2s−2)−(3s−4)=s−1
i,j k
and hence dim V ∩V =s−1 which is a contradiction.
i k
(cid:3)
Nowwearereadyforthemainresultofthissection. Itdescribesthecodimension
2 strata of discriminantal arrangements having multiplicity 3 and shows absence of
codimension 2 strata having multiplicity 4 (with obvious exceptions).
Theorem 3.8. Let A0 be generic arrangement of hyperplanes in Pk 1.
−
1. The arrangemen∞t B(n,k,A0 ) has (cid:0) n (cid:1) codimension 2 strata of multiplicity
k+2
k+2. ∞
2. There is one to one correspondence between
a) triples of dependent subspaces which are intersections of hyperplanes of re-
striction of a subarrangement of A0 onto an intersection of hyperplanes of A0
and ∞ ∞
b) triples of hyperplanes in B(n,k,A0 ) for which the codimension of their in-
tersection is equal to 2. ∞
(Recall that existence of dependent arrangement which is a restriction of a sub-
arrangement of A0 to intersection of hyperplanes in A0 mentioned in a) means
that there are hype∞rplanes H0,...,H0 in A0 such that k∞−1−t=2s−2 and there
are 3s hyperplanes in A0 ini1tersectiiotns of∞which with (cid:83)t H0 give arrangement
which is dependent in the∞sense of definition 3.2.) j=1 ij
3. There are no codimension 2 strata having multiplicity 4 unless k = 3. All
codimension 2 strata of B(n,k,A0 ) not listed in part 1, have multiplicity either 2
or 3 (the latter corresponding to t∞riples of hyperplanes in b).
4.Combinatorial type of B(n,2,A0 ) is independent of A0.
∞
Proof of Theorem 3.8. Thestatement(1)followsimmediatelyfromdiscussionafter
(6) in section 2.2. If J ⊂ {1,...,n} is a subset of cardinality k+2, then D is a
J
codimension 2 subspace in Cn and belongs to k+2 hyperplanes D ,K ⊂J.
K
NextweshalldeterminetheconditionsonthesetsK ,K ,K ,i(cid:54)=j (cid:54)=l (andthe
i j l
geometryofthecorrespondingstratainA0 )underwhichCodimD ∩D ∩D =
Ki Kj Kl
2. ∞
Consider first the case when sets K ,K ,K , each having cardinality k+1, are
i j l
such that for one of the them, say K , one has K \(K ∩(K ∪K )) (cid:54)= ∅,i (cid:54)=
i i i j l
j (cid:54)= l i.e. one of the set in this triple is not in the union of other two. The
dimension of the space of arrangements in D ∩D is n−2 since K (cid:54)= K . If
Kj Kl j l
r ∈K \(K ∩(K ∪K )),thenhyperplanesinanarrangementa∈D ∩D ∩D
i i j l Ki Kj Kl
with indices different from r or from indices in K ∩K can be chosen as arbitrary
j l
10 ANATOLYLIBGOBERANDSIMONASETTEPANELLA
parallel translates of hyperplanes in A0, while H is fixed by condition a being in
r
D . Hence D ∩D ∩D (cid:54)=D ∩D i.e. codimD ∩D ∩D =3.
Ki Ki Kj Kl Kj Kl Ki Kj Kl
From now on we shall assume that the triple K ,K ,K is such that
i j l
(cid:91)
(11) K =(K ∩K ) (K ∩K )
i i j i l
(cid:84)
aswellasforanypermutationof(i,j,k). NextletL =(K ∩K )\ K ,t=
α,β α β α=i,j,k α
(cid:84) (cid:84)
Card K ,l = CardL . Then (11) implies that K \ K =
α=i,j,k α α,β α,β β α=i,j,k α
(cid:83)
L L and since CardK =k+1 we have
α,β β,γ i
(12) l +l +t=k+1, α(cid:54)=β (cid:54)=γ
α,β β,γ
Using these relations for allowable permutations of subscripts, yields:
k+1−t
(13) l = α(cid:54)=β,α,β ∈{i,j,k}
α,β 2
ForatripleofsubsetsK ,K ,K ,CardK ∩K ∩K =tandthefixedarrangement
i j l i j l
A0 at infinity, consider the map
∞
S(H0,...,H0)→Ct =S(...,H0,...) r ∈K ∩K ∩K
1 n r i j l
having
S(...,H0 ∩( (cid:92) Htβ),...), α∈[1,...,n]\K ∩K ∩K
α β i j l
β∈Ki∩Kj∩Kl
as its fiber and which assignes to a collection of n parallel translates Ht1,...,Htn
1 n
of H0,...,H0 in Ck, the intersections of the hyperplanes with indices outside of
1 n
K ∩K ∩K with the linear subspace which is intersection of t hyperplanes with
i j l
indicesinK ∩K ∩K . Ifsisthedimensionofthefamilyofarrangementswhichis
i j l
intersection of hyperplanes D ,α=i,j,l then the dimension of induced on Ck t
Kα −
family is s−t. Hence
(14) CodimDKi∩DKj∩DKl =CodimDKi\(cid:84)Kα∩DKj\(cid:84)Kα∩DKl\(cid:84)Kα α=i,j,l
where the intersection on the right is taken in the space of parallel translates in
(cid:84)
jC∈(cid:84)leKarilHyjt.<k and in the case when t=k−1 we have l =1 i.e. Card(cid:83)K =
i,j i
k+2 and we are in the case (1) i.e. the codimension 2 stratum has multiplicity
k+2. Ift=0,t<k−1thenwehavethecaseconsideredinLemma3.1andwealso
have by this lemma that intersection of D ,i=1,2,3 has codimension two if and
Ki
only if the assumptions of the theorem are fulfilled. The rest of the part (2) of the
(cid:84)
theorem follows from lemma 3.4 for restriction on H and relation
(14) (with s=l ). α∈Ki∩Kj∩Kl α
α,β
Now consider the existence of codimension 2 strata of multiplicity 4. Sup-
pose that such stratum exists and K ,i = 1,...,4 are the corresponding subsets
i
of {1,...,n}. By quadruples analog of restriction (14) it is enough to consider
(cid:84)
the case K = ∅. Let l = CardK ∩ K ∩ K . Then for any i,
i=1,...,4 i i,j,m i j m
(cid:84) (cid:83)
CardK ∪ K = Card K i.e. l +k+1 is independent of (i,j,m) and
i j=i j i i,j,m
hence one infe(cid:54) rs from (12) the relation l = k+1.
(cid:84) i,j,m 3
Notethatcodim D =2ifandonlyifcodim D ∩D ∩D =2
i=1,...,4 Ki Ki1 Ki2 Ki3
for all 4 triple 1≤i ≤4 of distinct integers. Applying part (2) of the theorem to
j
each triple i ,i ,i one infers existence of quadruple of subspaces as in Lemma 3.7.
1 2 3
Hence this lemma implies the part (3).
