/^creative ^commor
ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 16 (2019) 257-276
https://doi.org/10.26493/1855-3974.1619.a03
(Also available at http://amc-journal.eu)
Pappus's Theorem in Grassmannian Gr (3, Cn)
Sumire Sawada, Simona Settepanella *, So Yamagata
Department of Mathematics, Hokkaido University, Japan
Received 26 February 2018, accepted 13 August 2018, published online 6 January 2019
Abstract
In this paper we study intersections of quadrics, components of the hypersurface in the Grassmannian Gr(3, Cn) introduced by S. Sawada, S. Settepanella and S. Yamagata in 2017. This lead to an alternative statement and proof of Pappus's Theorem retrieving Pappus's and Hesse configurations of lines as special points in the complex projective Grassmannian. This new connection is obtained through a third purely combinatorial object, the intersection lattice of Discriminantal arrangement.
Keywords: Discriminantal arrangements, intersection lattice, Grassmannian, Pappus's Theorem. Math. Subj. Class.: 52C35, 05B35, 14M15
1 Introduction
Pappus's hexagon Theorem, proved by Pappus of Alexandria in the fourth century A.D., began a long development in algebraic geometry.
In its changing expressions one can see reflected the changing concerns of the field, from synthetic geometry to projective plane curves to Riemann surfaces to the modern development of schemes and duality.
(D. Eisenbud, M. Green and J. Harris [4])
There are several knowns proofs of Pappus's Theorem including its generalizations such as Cayley Bacharach Theorem (see Chapter 1 of [9] for a collection of proofs of Pappus's Theorem and [4] for proofs and conjectures in higher dimension).
In this paper, by mean of recent results in [6] and [10], we connect Pappus's hexagon configuration to intersections of well defined quadrics in the Grassmannian providing a new statement and proof of Pappus's Theorem as an original result on dependency conditions for defining polynomials of those quadrics. This result enlightens a new connection
*The second named author was supported by JSPS Kakenhi Grant Number 26610001. E-mail addresses: b.lemon329@gmail.com (Sumire Sawada), s.settepanella@math.sci.hokudai.ac.jp (Simona Settepanella), so.yamagata.math@gmail.com (So Yamagata)
©® This work is licensed under https://creativecommons.org/licenses/by/4.0/
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between special configurations of points (lines) in the projective plane and hypersurfaces in the projective Grassmannian Gr(3, Cn). This connection is made through a third combinatorial object, the intersection lattice of the Discriminantal arrangement. Introduced by Manin and Schechtman in 1989, it is an arrangement of hyperplanes generalizing classical braid arrangement (cf. [7, p. 209]). Fixed a generic arrangement A = {H0,..., H} in Ck, the Discriminantal arrangement B(n, k, A), n, k G N for k > 2 (k = 1 corresponds to Braid arrangement), consists of parallel translates H^1,..., Htn, (ti,..., tn) G Cn, of A which fail to form a generic arrangement in Ck. The combinatorics of B(n, k, A) is known in the case of very generic arrangements, i.e. A belongs to an open Zariski set Z in the space of generic arrangements H0, i = 1,..., n (see [7], [1] and [2]), but still almost unknown for A G Z .In 2016, Libgober and Settepanella (cf. [6]) gave a sufficient geometric condition for an arrangement A not to be very generic, i.e. A G Z .In particular in the case k = 3, their result shows that multiplicity 3 codimension 2 intersections of hyperplanes in B(n, 3, A) appears if and only if collinearity conditions for points at infinity of lines, intersections of certain planes in A, are satisfied (Theorem 3.8 in [6]). More recently (see [10]) authors applied this result to show that points in a specific degree 2 hypersurface in the Grassmannian Gr(3, Cn) correspond to generic arrangements of n hyperplanes in C3 with associated discriminantal arrangement having intersections of multiplicity 3 in codi-mension 2 (Theorem 5.4 in [10]). In this paper we look at Pappus's configuration (see Figure 1) as a generic arrangement of 6 lines in P2 which intersection points satisfy certain collinearity conditions (see Figure 2). This allows us to apply results on [6] and [10] to restate and re-prove Pappus's Theorem.
More in details, let A be a generic arrangement in C3 and the arrangement of lines in ~ P2 directions at infinity of planes in A. The space of generic arrangements of n lines in (P2)n is Zariski open set U in the space of all arrangements of n lines in (P2)n. On the other hand in Gr(3, Cn) there is open set U' consisting of 3-spaces intersecting each coordinate hyperplane transversally (i.e. having dimension of intersection equal 2). One has also one set U in Hom(C3, Cn) consisting of embeddings with image transversal to coordinate hyperplanes and U/ GL(3) = U' and U/(C*)n = U. Hence generic arrangements in C3 can be regarded as points in Gr(3, Cn). Let {si < • • • < s6} C {1,..., n} be a set of indices of a generic arrangement A = {H0,..., H} in C3, ai the normal vectors of H0's and = det(a^ aj, a;). For any permutation a G S6 denote by [a] = {{ii,i2}, {i3,i4}, {i5,i6}}, ij = sCT(j), and by QCT the quadric in Gr(3,Cn) of equation ^i2i5i6 - Pi2i3i4A^ie = 0. The following theorem, equivalent to the Pappus's hexagon Theorem, holds.
Theorem 5.3 (Pappus's Theorem). For any disjoint classes [ai] and [a2], there exists a unique class [a3] disjoint from [ai] and [a2] such that {QCT1, QCT2, QCT3} is a Pappus configuration, i.e.
3
Qai1 n Qffi2 = p| Qffi
i=i
for any {ii, i2} C [3].
In the rest of the paper, we retrieve the Hesse configuration of lines studying intersections of six quadrics of the form Qa for opportunely chosen [a]. This lead to a better understanding of differences in the combinatorics of Discriminantal arrangement in the complex and real case. Indeed it turns out that this difference is connected with existence of the Hesse arrangement (see [8]) in P2 (C), but not in P2 (R).
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From above results it seems very likely that a deeper understanding of combinatorics of Discriminantal arrangements arising from non very generic arrangements of hyperplanes in Ck (i.e. A <// Z), could lead to new connections between higher dimensional special configurations of hyperplanes (points) in the projective space and Grassmannian. Vice versa, known results in algebraic geometry could help in understanding the combinatorics of Discriminantal arrangements in the non very generic case. Moreover we conjecture that regularity in the geometry of Discriminantal arrangement could lead to results on hyperplanes arrangements with high multiplicity intersections, e.g., in the case k = 3, line arrangements in P2 with high number of triple points (see Remark 6.6). This will be object of further studies.
The content of the paper is the following. In Section 2 we recall definition of Discrim-inantal arrangement from [7], basic notions on Grassmannian, and definitions and results from [10]. In Section 3 we provide an example of the case of 6 hyperplanes in C3. In Section 4 we define and study Pappus hypersurface. Section 5 contains Pappus's theorem in Gr (3, Cn) and its proof. In the last section we study intersections of higher numbers of quadrics and Hesse configuration.
2 Preliminaries
2.1	Discriminantal arrangement
Let H0, i = 1,..., n be a generic arrangement in Ck, k < n i.e. a collection of hyperplanes such that codim p| ieK \K\=p H0 = p. Space of parallel translates (H0,..., H0) (or simply when dependence on H0 is clear or not essential) is the space of n-tuples H,..., Hn such that either Hi n H0 = 0 or Hi = H0 for any i = 1,..., n. One can identify with n-dimensional affine space Cn in such a way that (H0,..., H0) corresponds to the origin. In particular, an ordering of hyperplanes in A determines the coordinate system in (see [6]).
We will use the compactification of Ck viewing it as Pk (C) \ endowed with collection of hyperplanes H0 which are projecive closures of affine hyperplanes H0. Condition of genericity is equivalent to |Ji H0 being a normal crossing divisor in Pk (C).
Given a generic arrangement A in Ck formed by hyperplanes Hi, i = 1,..., n the trace at infinity, denoted by ATO, is the arrangement formed by hyperplanes HTOji = H0 n in the space ~ Pk-i (C). The trace of an arrangement A determines the space of parallel translates S (as a subspace in the space of n-tuples of hyperplanes in Pk).
Fixed a generic arrangement A, consider the closed subset of S formed by those collections which fail to form a generic arrangement. This subset of S is a union of hyperplanes DL c S (see [7]). Each hyperplane DL corresponds to a subset L = {ii,..., ik+i} C [n] := {1,..., n} and it consists of n-tuples of translates of hyperplanes H0,..., H0 in which translates of H0,..., H0 fail to form a general position arrangement. The arrangement B(n, k, A) of hyperplanes DL is called Discriminantal arrangement and has been introduced by Manin and Schechtman in [7]. Notice that B(n, k, A) depends on the trace at infinity hence it is sometimes more properly denoted by B(n, k, ATO).
2.2	Good 3s-partitions
Given s > 2 and n > 3s, a good 3s-partition (see [10]) is a set T = {L i, L2, L3}, with Li subsets of [n] such that |Li| = 2s, |Li n Lj | = s (i = j), L i n L2 n L3 = 0 (in
particular | U Li| = 3s), i.e. L i = {i i,..., i2S}, L2 = {i i,..., is,i2S+i,... ,i3s}, L3 =
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{¿s + 1, . . . , ¿3s}.
Notice that given a generic arrangement A in C2s-1, subsets L define hyperplanes DLi in the Discriminantal arrangement B(n, 2s - 1, ATO). In this paper we are mainly interested in the case s = 2 corresponding to generic arrangements in C3.
2.3	Matrices A () and At ()
Let a = (a^,..., ajk) be the normal vectors of hyperplanes ffj, 1 < i < n, in the generic arrangement A in Ck. Normal here is intended with respect to the usual dot product
(ai, .. ., afc) • (vi,. .., vfc) = ajVj.
j
Then the normal vectors to hyperplanes DL, L = {s1 < ••• < sfc+1} c [n] in S ~ Cn are nonzero vectors of the form
k+1
«L = 53(-1)j det(asi,. .. ,aSi,.. ., aSfc+i )eSi,	(2.1)
i=1
where {ej}1<j<n is the standard basis of Cn (cf. [2]).
Let Pk+1([n]) = {L c [n] | |L| = k +1} be the set of cardinality k +1 subsets of [n]. Following [10] we denote by
A(A^) = («L)L£Pfc+ i([n])
the matrix having in each row the entries of vectors aL normal to hyperplanes DL and by (Ato) the submatrix of A(ATO) with rows aL, L G T, T c Pk+1([n]). In this paper we are mainly interested in the matrix AT(ATO) in the case of T good 6-partition.
2.4	Grassmannian Gr(k, Cn)
Let Gr(k, Cn) be the Grassmannian of k-dimensional subspaces of Cn and
k
Y: Gr(k, Cn) ^ P(\Cn) («1,. .. ,vfc} ^ [v1 A • • • A Vk ],
the Plucker embedding. Then [x] G P(/\k Cn) is in y( Gr (k, Cn)) if and only if the map
k+1
: cn ^ \ Cn v ^ x A v
has kernel of dimension k, i.e. ker = (v1,..., vk}. If e1,..., en is a basis of Cn then
e/ = ej1 A • • • A eifc, I = {i1,..., ik} C [n], i1 < • • • < ik, is a basis for /\k Cn and k
x G /\ Cn can be written uniquely as
x = ^ e/ = XI ^¿l...ifc (eil A-"A eifc )
/C[n]	1<ii<---<jfc<n
|/| = k
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where homogeneous coordinates ,0/ are the Plucker coordinates on P(Ak cn) ~ p(k)-1(c) associated to the ordered basis e1,...,en of Cn. With this choice of basis for Cn the matrix Mx associated to is a (k+ J x n matrix with rows indexed by subsets I = {i1,...,ik }c [n] and entries
=
_ J (-l)1^/\{j> if j = ii e I,
0	otherwise.
Plucker relations, i.e. conditions for dim(ker y>x) = k, are vanishing conditions of all (n — k + 1) x (n — k +1) minors of Mx. It is well known (see for instance [5]) that Plucker relations are degree 2 relations and they can also be written as
k
E(—1)1 ..pk-iqi ftqo...qi...qk =
0 (2.2)
1=0
for any 2k-tuple (pi,... ,Pk-1, qo,..., qk).
Remark 2.1. Notice that vectors in the equation (2.1) normal to hyperplanes correspond to rows indexed by L in the Plucker matrix Mx, that is
A(ATO ) = Mx,
up to permutation of rows. Notice that, in particular, det(asi,..., ofSi,..., aSfc+1) is the Plucker coordinate ,0/, I = {s1, s2,..., sk+1} \ {«»}.
2.5 Relation between intersections of lines in and quadrics in Gr (3, Cn)
Let A = {H0,..., H} be a generic arrangement in C3. If there exist L1, L2, L3 c [n] subsets of indices of cardinality 4, such that codimension of DLl n DL2 n DL3 is 2 then A is non very generic arrangement (see [2]).
Let T = {L1, L2, L3} be a good 6-partition of indices {s1,...,s6} C [n]. In [6], authors proved that the codimension of DLl n DL2 n DL3 is 2 if and only if points
f1ieL1nL2	f1ieL1nL3	and PlieL2nL3
are collinear in ([6, Lemma 3.1]).
Since is vector normal to , the codimension of DLl n DL2 n DL3 is 2 if and only if rank AT(ATO) = 2, i.e. all 3 x 3 minors of AT(ATO) vanish. In [10] authors proved the following Lemma.
Lemma 2.2 ([10, Lemma 5.3]). Let A be an arrangement of n hyperplanes in C3 and
O.T = {{i1, i2, i3, i4}, {i1,i2, i5,i6}, {i3, i4, i5, i6}}
a good 6-partition of indices s1 < • • • < s6 e [n] such that j = sCT(j), o permutation in S6. Then rank ACT.T(ATO) = 2 if and only if A is a point in the quadric of Grassmannian Gr (3, Cn) of equation
£¿2 ¿5 ¿6 ^¿2«3i4 ^¿1i5«6
0.	(2.3)
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As consequence of above results, we obtain correspondence between points
x = ^ Piej, pj = 0,
IC[n] | I | =3
in the quadric of equation (2.3) and generic arrangements of n hyperplanes A in C3 such that n HOTji2, HOTjia n and HOTji5 n HOTji6 are collinear in HOT. Notice that condition pI = 0 is direct consequence of A being generic arrangement.
3 Motivating example of Pappus's Theorem for quadrics in Gr (3, Cn)
In classical projective geometry the following theorem is known as Pappus's theorem or Pappus's hexagon theorem.
Theorem 3.1 (Pappus). On a projective plane, consider two lines l1 and l2, and a couple of triple points A, B, C and A', B', C' which are on 11 and l2 respectively. Let X, Y, Z be points of AB' n A'B, AC' n A'C and BC' n B'C respectively. Then there exists a line l3 passing through the three points X, Y, Z (see Figure 1).
Figure 1: Original Pappus's Theorem.
This theorem was originally stated by Pappus of Alexandria around 290-350 A.D.
In this section, we restate this classical theorem in terms of quadrics in the Grassman-nian. Indeed the six lines AB', A'B, BC', B'C, AC', A'C e P2 (C) correspond to lines in the trace at infinity of a generic arrangement A in C3 and lines l1, l2 and l3 correspond to collinearity conditions for intersection points of lines in AOT.
Consider a generic arrangement A = {H1,..., H6} of 6 hyperplanes in C3, its trace at infinity and T = {L1,L2,L3} the good 6-partition defined by L1 = {1,2,3,4}, L2 = {1,2,5,6}, L3 = {3,4,5,6}. By Lemma 2.2 we get that the triple points
nieLinL2 Hi n H~, fWLa Hi n H~, nieL2nL3 Hi n H~
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are collinear if and only if A is a point of the quadric
Ql : ^134^256 — ^234^156 = 0
in Gr(3, C6). Analogously if
T' = {Ll, L2, ¿3}, Ll = {4, 6, 2, 5}, ¿2 = I4, 6,1, 3}, ¿3 = {2, 5,1, 3}
and
T'' = {¿1', ¿2', ¿3'}, ¿1' = {2,4,1, 6}, ¿2' = {2, 4, 3, 5}, ¿3' = {1, 6, 3, 5} are different good 6-partitions then triple points
fliGLJnLi. Hi n nieiinL3 Hi n HieL^nL^ ^ n
and
PlieL'/nL^' H n	PlieL'/nL^' Hi n	PlieL2'nL3' Hi n
are collinear if and only if A is, respectively, a point of quadrics
Q2: ^425^613 — ^625^413 = 0 and Q3 : ^216^435 — ^416^235 = 0.
With above remarks and notations we can restate Pappus's Theorem as follows (see Figure 2).
Theorem 3.2 (Pappus's Theorem). Let A = {H1,..., H6} be a generic arrangement of hyperplanes in C3. If A is a point of two of three quadrics Q1, Q2 and Q3 in the Grassmannian Gr (3, C6), then A is also a point of the third. In other words
3
Qi ' n Qi2 = ff Qi, {¿1,i2}c [3].
i=1
We develop this argument in the following sections providing in Theorem 5.3 a general statement on quadrics in the Grassmannian which implies Pappus hexagon Theorem in the projective plane.
4 Pappus Variety
In this section, we consider a generic arrangement {H1,..., Hn} in C3 (n > 6). Let's introduce basic notations that we will use in the rest of the paper.
Notation. Let {s1,..., s6} be a subset of indices {1,..., n} and T = {¿1, ¿2, ¿3} be the good 6-partition given by
¿1 = {s1, S2, S3, S4}, ¿2 = {s1, S2, S5, S6} and ¿3 = {S3, S4, S5, S6}.
Then for any permutation a G S6 we denote by a.T =	, aX3} the good 6-
partition given by subsets
= {¿1, ¿2, ¿3, ¿4}, a^2 = {¿1, ¿2, ¿5, ¿6} and 0X3 = {¿3, ¿4, ¿5, ¿6} with j = sCT(j). Accordingly, we denote by QCT the quadric in Gr (3, Cn) of equation
Q : A 1 i3 i4 ^i2i5i6 ^i2i3i4 Ali5i6 0.
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The following lemma holds.
Lemma 4.1. Let ct, ct' g S6 be distinct permutations, then QCT = if and only if there exists t g S3 such that ct.Lj n CT.Lj = ct'.Lt(j) n ct'.Ltj (1 < i < j < 3).
Proof. By definition of good 6-partition we have that
Li = (Li n L2) U (Li n L3), L2 = (L2 n Li) u (L2 n L3), L3 = (L3 n Li) u (L3 n L2).
Then there exists t g S3 such that ct and ct' satisfy a.Lj n a.Lj = ct'.Lt(^ n ct'.Ltj (1 < i < j < 3) if and only if ct.L; = ct'.Lt(() for l = 1,2,3, that is ACT/.T(AOT) is obtained by permuting rows of ACT.T(AOT). It follows that rank ACT.T(AOT) = 2 if and only if rank ACT/.T(AOT) = 2 and hence by Lemma 2.2 this is equivalent to QCT n NS1... S6 = Qct' n NS1i...jS6, where
NS1I...IS6 = {x = E ei I = 0 for any I c {si,... ,se}}.
IC[n]
|I|=3
Since NS1 i...jS6 is dense open set in 7(Gr(3,Cn)), Q- n NS1i...iS6 = Q- n NS1i...jS6 if and only if QCT = . Vice versa if QCT n Ns1... s6 = n Ns1... s6, then any generic arrangement A corresponding to a point in QCT n Ns1. . . s6 corresponds to a point in n NS1 i...jS6, that is rank ACT.T(AOT) = 2 if and only if rank ACT/.T(AOT) = 2. It follows that Act T(Aot) and ACT/.T(AOT) are submatrices of A(AOT) defined by the same three rows, i.e. ct.L; = ct'.Ltfor l = 1,2,3.	□
Definition 4.2. For any 6 fixed indices T = {si,..., s6} C [n] the Pappus Variety is the hypersurface in Gr(3, Cn) given by
pt = U Q-.
CTGS6
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Notice that all the content of this section and the following section is based on the choice of six indices {si < • • • < s6} C [n]. This is related to result in Theorem 3.8 in [6] and, consequently, Lemma 5.3 in [10] (Lemma 2.2 in this paper). Indeed Theorem 3.8 in [6] states that in order to study special configurations of n lines in P2, that is non very generic arrangements of n lines in P2, it is sufficient to study subsets of six lines out of n. On the other hand since Pappus Variety can be defined inside Gr(3, Cn), we decided to keep the discussion more general picking six indices {s1 < ••• <s6}C [n] instead of simply study the case Gr (3, C6) (see also Remark 6.7).
For a, a' G S6 we define the equivalence relation a.T ~ a'.T corresponding to QCT = Qa> as following:
a.T - a'.T t g S3 such that a.L, n a.Lj = a'.LT(i) n a'.LT(1 < i < j < 3).
We denote by [a] the equivalence class containing a.T and by QCT the corresponding quadric (notice that a in the notation QCT can be any representative of [a]). By Lemma 4.1 [a] only depends on couples Lj n Lj hence for each class [a] we can choice a representative
a.To = {{ji, j2, j3, j'4}, {ji, j2, j5, je}, {j3,j4,j5, je}} such that j < j2, j3 < j4, j5 < j6 and j < j'3 < j5 and we can equivalently define
[a] = {{j1, j2}, {j3, j4}, {j5,j6}}. (6)(4)(2)
Since the number of choices of [a] is V2y v32/V2y =15, Pappus Variety is composed by 15 quadrics. Finally remark that
[a] = {{ji, j'2}, {j'3, j'4 }, {j5, j'e}} and [a'] = {{ji,j2}, {j'3,j'4}, {j'5,j'6}}
are disjoint, i.e. [a] n [a'] = 0, if and only if {j2;-i, j'2;} = {j'2i'-i,j'2v} for any 1 < /,/' < 3.
Definition 4.3 (Pappus configuration). Let [ai], [a2] and [a3] be disjoint classes, a Pappus configuration is a set {QCT1, QCT2, QCT3} of quadrics in Gr (3, Cn) such that
3
Q<xn n Qffi2 = PI Qffi i=i
for any {ii,i2} C [3].
Quadrics Qai, QCT2, QCT3 are said to be in Pappus configuration if {QCT1, QCT2, QCT3} is a Pappus configuration.
Remark 4.4. Fixed a class of good 6-partition [a] = {{ji,j2}, {j3,j4}, {j5,j6}}, we shall count the number of disjoint classes. {	}
First let's count the number of classes [a'] = { {ji, j2}, {j3, j'4 }, {j'5, j'6 } } not disjoint and distinct from [a]. Since [a] and [a'] are distinct, only one couple {j', j'+i} is contained in [a]. Without lost of generality we can assume {j;, jl+i} = {j', j'2} (/ is either 1,3 or 5) then pairs {j3, j'4} and {j5, j'6} are not in the same set, i.e. we have two possibilities:
{j'3,j'5} and {j4,j6} G [a],
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or
{jj} and j}e [a].
Hence there are 2 • 3 + 1 =7 not disjoint classes from [a] and, since the number of all classes is 15, we get that any fixed [a] admits exactly 15 - 7 = 8 disjoint classes.
5 Pappus's Theorem
In this section we restate Pappus's Theorem for quadrics in Gr(3, Cn) by using notation introduced in the previous section. For a fixed class [a] = {{ji, j2}, {j3,j4}, {js,j6}} let's denote by G[CT] the free group generated by permutations of elements in each subset of [a], that is
gm = (fe-i j2i) e S6 | l =1,2, 3), and, for any class, [a'] let's define the set
orbitGM([a']) = {t[a'] | t e GM}
where t acts naturally as permutation of entries of each set in [a'].
Remark 5.1. The action of G[ctj on class [a'] disjoint from [a] is faithful. Indeed let
t, t' g Gm be such that t[a'] = r'[a'] then t-1t'[a'] = [a'], i.e. t-1t' G G[ct,j. Thus we get t-1t' G G[ctj n G[ff/j. Since [a] and [a'] are disjoint, G[CT] n G[ff/j = {e}, i.e., t = t'. Remark that | orbit G ([a'])| = |G[CT]| = 8 and t [a] = [a] for any t g G[ctj.
Lemma 5.2. Let [a] and [a'] be disjoint classes, then
orbitGw([a']) = {[a''] | [a] n [a''] = 0}.
Proof. First we prove that orbit g w ([a']) c {[a''] | [a] n [a''] = 0}. Let
[a] = {{ji, j2}, {j3, j4}, {j5, j6}} and [a'] = {{jl,j2}, {j'3,j4}, {j'5,j6}}
be disjoint, then |{j2i_i, j2i} n {j2m_i, j'2m}| < 1. Since t g G[ct] permutes only j2i-i and j2l then t [a']n[a] = 0, that is t [a'] is disjoint from [a], i.e. t [a'] G {[a''] | [a]n[a''] = 0{. Since G[ctj is faithful, | orbitGw ([a'])| = 8 and, by calculations in the Remark 4.4, |{[a''] | [a] n [a''] = 0}| = 8, it follows that orbitGw ([a']) = {[a''] | [a] n [a''] = 0}. □
The following theorem holds.
Theorem 5.3 (Pappus's Theorem). For any disjoint classes [a] and [a'], there exists a unique class [a''] disjoint from [a] and [a'] such that {QCT, , } is a Pappus configuration.
Remark 5.4. Let [ai] and [a2], [a»] = {{ji,i, j2,i}, {j3,i,j4,i}, {j5,i, j6,i}}, i = 1,2 be classes of indices in {1,..., 6}. Recall the following facts (see Section 2 and Lemma 2.2):
i) If
x = E & ei, Pi = 0
IC[6] HI =3
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is a point in QCTi then any arrangement A G C3 such that A(Aœ) = Mx is an arrangement of 6 planes in general position in C3 with lines in such that points
n H^,32,i7 H^,33,i n H^,34,i and	n
are collinear.
ii) Vice versa if A is an arrangement of 6 lines in general position in C3 with the intersection points
Hœ,ji,i n H^,32,i 7 H^,33,i n	and Hl^,35,i n
collinear, then any point
x = Pi ei
IC[6]
|I|=3
such that Mx = A(A«) verifies pI = 0 and x g QCTi.
From ii) it follows that if is an arrangement of 6 lines in general position in P2 such that
Hœ,ji,i n	,i 7	,i n H^,34,i and Hœ,j5,i n Hœ,j6,i
are collinear for i = 1,2, then any point
E Piei
IC[6]
|I|=3
such that Mx = A(A«) belongs to QCTl n QCT2. Moreover [ai] and [a2] are disjoint classes. By Theorem 5.3 there exists a third class
[03] = {{ji,3, j2,3}, {j3,3, j'4,3}, {j5,3,j6,3}
such that {QCTl, QCT2, QCT3} is a Pappus configuration. Then x G p| 1<;3 QCTi which implies, by i) that also
Hœ,ji,3 n Hœ,j2,37 H^,33,3 n H^,J4,3 and Hœ,j5,3 n Hœ,j6,3
have to be collinear. That is Theorem 5.3 implies Pappus hexagon Theorem in the plane (see Figure 2).
Notice that Theorem 5.3 is slightly more general than Pappus hexagon Theorem since it also applies to the case in which some pI = 0.
Proof of Theorem 5.3. Following example in Section 3, for any class
M = {{j1,j2}, {j3, j4}, {j5, j6}}
let's consider disjoint classes
M = {{j1 ,j3}, {j2 ,j5}, {j4, j6}} and M = {{j1,j6}, {j2,j4}, {j3, j5}}.
x=
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The corresponding quadrics have equations:
Qwi
QW2 QU3
Pjij3j4 Pj2j5 j6 — Pj2j3j4 Pjij5j6 — 0, Pj4j2j5 Pj6ji j3 — Pj6j2j5 Pj4jij3 — 0, Pj5jij6 Pj3j2 j4 — Pj3jij6 Pj5j2j4 — 0
By definition of equations of QW2 and QW3 can equivalently be written as
QU2 : Pj2j4j5 Pjij3 j6 + Pj2j5j6 Pjij3j4 — 0, QU3 : Pjij5j6 Pj2j3 j4 + Pjij3j6 Pj2j4j5 — 0
If we denote left side of defining equations of QWi by PWi then
P - P — P
J W2 J Wi	J W3
that is zeros of any two polynomials PWi , PWi2 are zeros of PWi , {¿i, ¿2, ¿3} — {1, 2,3}. We get i 2 3
3
Qwi1 n Qwi2 — H Qwi ¿=1
for any {¿1, i2} c [3], i.e. QWi, QW2 and QW3 are in Pappus configuration.
By Lemma 5.2, since [w1] n [w2] — 0, the set of disjoint classes from [w1] is given by
{[ao] | M n [ao] — 0} — {toH | to e G^]}.
Then if [a'] is disjoint from [w1 ], there exists a unique element t e G[Wi] such that [a'] — t[w2]. That is, for a generic class [w1], any disjoint couple ([w1], [a']) is of the form ([w1], t[w2]) — (t[w1], t[w2]) and we have
QW1 — QTW1 : ^r(ji )T(j3 )T(j4)PT( j2 )t(j5 )t(j6 ) PT(j2)T(j3 )t(j4)PT(ji )t(j5)T (j6 ) 0, Q<j' — QTW2 : PT(j4 ) T(j2 )T(j5)PT(j6 )t(ji )t(j'3) PT(j6 )t(j2 )t(j5 )PT ( j4 )t(ji )t(j3 )
By antisymmetric property of indices of ^¿jfc, if we denote by PWi and PCT/ the left side of above equations, i.e.
Pw1 — pt (ji)T (j3 )t (j4)Pt (j2 )t (j5 )t (j6) — pt (j2 )t (j3 )t (j4)Pt (ji )t (j5)T (j6 ) , p<t' — PT (j4 )t(j2 )t(j5 )PT ( j6 )t(ji )t(j3 ) - PT(j6 )t(j2 )t(j5) pt(j4 ) T(ji )t(j3 )
then
Pt" :— Pt' — Pwi — PT (j5 )t (ji )t (j6 ) PT (j3 )t (j2)T (j'4 ) — PT j^T (ji )t (j6 ) PT (j5 )t (j2)T (j'4 )
is the defining polynomial of QTW3. That is [a''] is uniquely determined by disjoint couple
(["1], [a']).	□
From proof of Theorem 5.3 we get that for any class
["1] — {{j1,j2}, {j3, j4}, {j5, j6}}
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if we denote
M = {{j1 Jsh { j2, j5 }, { j4, j6 } } and [w3] = {O'ljeh { j2, j4 }, fe j5}},
then all Pappus configurations are of the form {QTW1, QTW2, QTW3}, t g G[W1] and the following Corollary holds.
Notice that the proof of Theorem 5.3 only uses equations of quadrics QCT and hence provides alternative proof to Pappus hexagon Theorem. In particular it is also alternative to classical proof based on Grassmann-Plucker relations. Indeed the latter proof uses the fact that points in Pappus configurations verify the Grassmann-Plucker relations while, in our cases, quadrics QCT are proper quadrics in the Grassmannian, i.e. equations of quadrics Qct are not Grassmann-Plucker relations.
Corollary 5.5. The number of Pappus configurations {QCT, Qff/, QCT" } in Gr (3, C6 ) is 20.
Proof. By Remark 4.4 the number of [a] is 15 and by Lemma 5.2 each fixed class [a] admits 8 disjoint classes. By Theorem 5.3 if [a] and [a'] are fixed, [a"] is uniquely determined, thus the number of the sets {[a], [a'], [a'']} is 15 x 8/3! = 20.	□
Corollary 5.5 establishes that for any given 6 lines in P2 there are 20 possible combinations of their intersections that give rise to a Pappus's configuration like the one in Figure 2.
6 Intersections of quadrics
In this section we study intersections of quadrics in Gr(3, Cn). In particular we are interested in the intersection of sets
Q° = Qff n {x = ^ 0/e/ | = 0 forany I C {si,..., s6}}
1C[n]
|1|=3
of points in quadrics QCT that correspond to arrangements of lines in P2 (C) with subarrangement {HS1,..., Hs6} generic. The following lemma holds.
Lemma 6.1. If [a1], [a2], [a3] are distinct andpairwise not disjoint classes then
Q°i n q°2 n q°3 = 0. Proof. If [a1], [a2], [a3] are not disjoint then either
(1)	|[ai] n [a2] n [a3]| = 1 or
(2)	|[aii] n [a<2]| = 1(1 < ii < i2 < 3) and [ai] n [a2] n [a3] = 0.
(1) Assume [ai] n [a2] n [a3] = {¿1 ,¿2}. Let [ai] = {{¿1, ¿2}, {¿3, ¿4}, {¿5, ¿6}}, [a2] = {{¿1, ¿2}, {¿3, ¿5}, {¿4, ¿6}}, and [a3] = {{¿1, ¿2}, {¿3, ¿6}, {¿4, ¿5}} then we obtain the following quadrics
Q<ri
Q.2
Q.3
¿5¿6	0Î2Î3Î4 0Î1Î5Î6	0,
0Î1Î3Î5 0Î2Î4¿6	— 0i2i3i5 ^¿^¿6	= 0, 0Î1Î3Î6
¿4¿5	0i2i3i6 0i1i4i5	0.
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Any point x G Q°1 n Q°2 belongs to Gr(3, Cn), that is x satisfies Plucker relations in (2.2). In particular x G Pli n Pl2 where Pli and Pl2 are the quadrics:
PI i : £j 1Î3Î2£Î4Î5Î6 £Î1Î3Î4 £¿2 Î5»6 + £ ¿1¿3¿5^¿2¿4¿6	¿3¿6 £¿2 ¿4 ¿5
Pl2 : £Î2Î3»1 £Î4¿5¿6 — £Î2¿3¿4£Î1Î5»6 + £Î2»3Î5 £Î1Î4»6 — £Î2»3Î6 £Î1Î4»5
Notice that Pl i and Pl2 can be obtained from equations in (2.2) considering the 6-tuples
(pi,p2, qo, qi, q2, 93) = (¿i, ¿3, ¿2, ¿4, ¿5, ¿6) and (¿2, ¿3, ¿i, ¿4, ¿5, ¿6) respectively. We get
Q.2 - Q CT1 Pl i + Pl2 : £Î1 ¿3 ¿6 £Î2¿4¿5 £î2î3î6 £Î1Î4Î5 + 2(£ ¿1 ¿2¿3 £î4Î5Î6 )
Since £i1i2Î3 = 0 and £¿44516 = 0 then £¿14243£î4î5î6 = 0 and hence
A1Î3Î6 £Î2Î4Î5 — £¿2¿3¿6 £Î1Î4Î5 = 0,
that is x G Qct3 .
(2) Assume [ai] n [02] = {¿i, ¿2}, [ai] n [03] = {¿3, ¿4} and [02] n [03] = {¿5, ¿6} and name Pi = {¿i, ¿2}, P2 = {¿3, ¿4}, P3 = {¿5, ¿6}. To any point x G Q°1 n Q°2 n Q°3 corresponds the existence of an arrangement with a generic sub-arrangement indexed by {¿i,..., ¿6} which trace at infinity {HTOjj1,..., } satisfyies collinearity conditions as in Figure 3. That is there exist couples P4 G [ai],P5 G [a2] and P6 G [a3] that correspond, respectively, to intersection points p4, p5 and p6 of lines in {HTOjj1,..., } (see Figure 3).
Figure 3: Case (2) trace at infinity of A G p|3=1 Q°., {i, j} corresponds to n .
By definition of P1; P2 and P3 we have
P3 = {i5, i6} G ({ii, . . . , ie} \ Pl) n ({ii, . . . , ie} \ P2).
On the other hand, if P4 is different from P1 and P2 in Q°1 then P4 = ({i1,..., i6}\P^ n ({i1,..., i6} \ P2). Thus we get P3 = P4 and, similarly, P5 = P2 and P6 = P^ that is Q° = Q° = Q° which contradict hypothesis.	□
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Lemma 6.2. For any three pairwise disjoint classes [<ri], [72], [<73], either |QCT1, Q-2, Q-3} is a Pappus configuration or
n q-=0.
¿=1
Proof. By Pappus's Theorem, for any two disjoint classes [<i], [<j], there exists [<ij] such that {QCTi, Qaj, Qaij} is Pappus configuration. If [<ij] = [<k] for some k G [3], then |Qff1, Q-2, Q-3} is a Pappus configuration. Thus assume all [<ij] = [<k] for any k = 1,2,3. Moreover [<12], [<13], [<23] are distinct since if [<ij] = [<ik] then <] = [<k]. If [712] n [713] = 0, [712] n [<23] = 0 and [713] n [723] = 0, then
H Q-n, = 0
1<Î1<Î2<3
by Lemma 6.1 and
33
H Q- = (H Q-) n ( H Q^ ) =0.
¿=1	¿=1	1 < 11 <12 <3
Otherwise assume [712] n [713] = 0, we get a new Pappus configuration. Since the number of disjoint classes is finite, iterating the process, we will eventually get 3 classes K ], [7l2], [7l3] pairwise not disjoint and
33
H Q- = (H Q-i) n n Q-i2 n Q-i3 = 0.	□
¿=1 ¿=1
Lemma 6.3. If [a1], [72], [a3] are distinct classes such that [a1] n [72] = 0 and K] n [73] = 0 for i = 1,2, then
3
H Q-i = 0.
¿=1
Proof. Since [a1], [a3] and [72], [a3] are disjoint, there exist [74] and [75] such that {Q-1, Q-3, Q-4} and {Q-2, Q-3, Q-5} are Pappus configurations and
[71] n [75] = 0, [72] n [74] = 0, [74] n [75] = 0.
Indeed if one of them is empty, we obtain 3 disjoint classes not in Pappus configuration and by Lemma 6.2, it follows
35
H Q-i = H Q-i = 0.
¿=1	¿=1
Since [71] n [72] = 0, we can assume {i1, i2} = [71] n [72] and we can set
[71]	= {{i1,i2}, {i3,i4}, {i5,i6^,
[72]	= {{i1,i2}, {i3,i4}, {i5,i6}},
[73]	= {i^^L {j3, j4}, j6}} .
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To any point
3
x g n q°i=0
i=1
corresponds an arrangement A with generic subarrangement {Hil,..., Hi6} with trace at infinity {HTOjil,..., HTO i6} intersecting as in Figures 4 and 5 (up to rename). It follows
Figure 4: Each j, j' is j or j2.
that {j4, je} € [04] and since {j'3, j'5} = {¿1, ¿2} € [01] and [01] n [04] = 0 (see Figure 4), there are two possibilities:
[o4] = {{j4, je}, Ob j3} {j2, j5}}
or
[04]	= { {j4, j6}, {j1, j5 }, {j2, j3 } }.
Analogously (see Figure 5) class [o5] is of the form
[o5] = {{j4, j6}, {j1, j3}, {j2, j5} }
or
[05]	= {{j4,je}, {j1,j5}, {j2,j3}}.
Since [01] n [0-5] = 0 and [0-5] ^ {j3, j'5} = {¿1, ¿2}, we deduce that j je} = {¿3, ¿4} or {¿5, ¿e}, which is not possible by [o1] n [o4] = 0. Hence
3
n Q-i=0.	°
¿=1
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Notice that the Hesse arrangement in P2 (C) (see Figure 6) can be regarded as a generic arrangement of 6 lines which intersection points satisfy 6 collinearity conditions.
Definition 6.4 (Hesse configuration). Let ^¿], 1 < i < 6 be distinct classes, we call Hesse configuration a set |QCT1,..., Q-6} of quadrics in Gr(3, Cn) such that there exist disjoint sets I, J C [6], |11 = | J| =3 such that {QCTi}ie1, {QCTj}jeJ are Pappus configurations and [a] n [aj j = 0 for any i G I, j G J.
With above notations, the following classification Theorem holds.
Theorem 6.5. For any choice of indices {si,..., s6} C [n] sets Q-, a G S6, in the Grassmannian Gr (3, Cn) intersect as follows.
(1)	For any disjoint classes [a1] and [a2], there exist [a3],..., [a6] such that {QCT1,..., Q-6} is an Hesse configuration for I = {1,2,3}, J = {4, 5,6} and
2	3	4	6
n q-=n Q- 2 n Q- 2 n Q- 2 0.
¿=1 ¿=1 ¿=1 ¿=1
(2)	For any not disjoint classes [a1] and [a2], there exist [a3],..., [a6] such that {QCT1,..., Q-6} is an Hesse configuration for I = {1,3,4}, J = {2, 5,6} and
2346
n q- 2 n q-=n q- 2 n q- 2 0.
¿=1 ¿=1 ¿=1 ¿=1
All other intersections are empty.
Remark 6.6. Notice that, since Hesse configuration only exists in the complex case, in Gr(3, Cn) we can find 6 quadrics {QCT1,..., Q-6} such that
6
n Q-i 2 0,
¿=1
while in Gr(3, Rn),
n Q-j=0.
je Jc[6]
|J|>4
It follows that in the real case, for any choice of indices {s1,..., s6} C [n], we have at most 4 collinearity conditions (see Figure 7) corresponding to 15 hyperplanes in the Dis-criminantal arrangement with 4 multiplicity 3 intersections in codimension 2 (see Figure 8). While in the complex case Hesse configuration (see Figure 6) gives rise to a Discriminantal arrangement containing 15 hyperplanes intersecting in 6 multiplicity 3 spaces in codimen-sion 2.
This remark allows a better understanding of differences in the combinatorics of Dis-criminantal arrangement in the real and complex cases. Indeed the existence of a discrimi-nantal arrangement of 15 hyperplanes intersecting in 6 multiplicity 3 spaces in codimension 2 in C but not in R implies that there exist combinatorics of Discriminantal arrangements that cannot be realised in any field. This is especially interesting since in the case known until now, i.e. in the case of very generic arrangements A, the combinatorics of Discriminantal arrangement B(n, k, A) is independent from the field (see [1]).
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Q,
Figure 7: Generic arrangement A in R3 containing 6 lines satisfying 4 collinearity conditions.
A "	"
Figure 8: Codimension 2 intersections of 15 hyperplanes in B(n, 3, ATO) indexed in {si,...,s6} C [n] with 4 multiplicity 3 points ▲ corresponding to intersections n3=i DCT.Li, f|3=i DCT/.Li, n3=i Dff//.Li and P|3=i DCT///.Li, ct, ct', a", a"' as in Figure 7.
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Remark 6.7. Finally Theorem 6.5 implies that the maximum number of intersections of multiplicity 3 in codimension 2 in the complex case is strictly higher than the one in the real case. This agrees with results on maximum number of triple points in an arrangement of lines in P2 (see [3] for a discussion on line arrangements with maximal number of triple points over arbitrary fields). Those observations suggest that special configurations of lines in the projective plane intersecting in a big number of triple points could be understood by studying Discriminantal arrangements with maximum number of multiplicity 3 intersections in codimension 2. Indeed each multiplicity 3 intersection in codimension 2 of B(n, 3, ) corresponds to a collinearity condition for lines in which is equivalent to the possibility to add a line that gives rise to "higher" number of triple points. It seems hence interesting to study exact number of intersections of type (1) and (2) in Theorem 6.5 in the Grassmannian Gr(3, Cn). This will be object of further studies.
References
[1]	C. A. Athanasiadis, The largest intersection lattice of a discriminantal arrangement, Beiträge Algebra Geom. 40 (1999), 283-289, https://www.emis.de/journals/BAG/vol. 40/no.2/1.html.
[2]	M. M. Bayer and K. A. Brandt, Discriminantal arrangements, fiber polytopes and formality, J. Algebraic Combin. 6 (1997), 229-246, doi:10.1023/a:1008601810383.
[3]	M. Dumnicki, L. Farnik, A. Glöwka, M. Lampa-Baczynska, G. Malara, T. Szemberg, J. Szpond and H. Tutaj-Gasinska, Line arrangements with the maximal number of triple points, Geom. Dedicata 180 (2016), 69-83, doi:10.1007/s10711-015-0091-7.
[4]	D. Eisenbud, M. Green and J. Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N. S.) 33 (1996), 295-324, doi:10.1090/s0273-0979-96-00666-0.
[5]	J. Harris, Algebraic Geometry: A First Course, volume 133 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1992, doi:10.1007/978-1-4757-2189-8.
[6]	A. Libgober and S. Settepanella, Strata of discriminantal arrangements, J. Singul. 18 (2018), 440-454, doi:10.5427/jsing.2018.18w.
[7]	Yu. I. Manin and V. V. Schechtman, Arrangements of hyperplanes, higher braid groups and higher Bruhat orders, in: J. Coates, R. Greenberg, B. Mazur and I. Satake (eds.), Algebraic Number Theory, Academic Press, Boston, MA, volume 17 of Advanced Studies in Pure Mathematics, pp. 289-308, 1989, in honor of K. Iwasawa on the occasion of his 70th birthday on September 11, 1987.
[8]	P. Orlik and H. Terao, Arrangements of Hyperplanes, volume 300 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1992, doi:10.1007/978-3-662-02772-1.
[9]	J. Richter-Gebert, Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, Springer, Heidelberg, 2011, doi:10.1007/978-3-642-17286-1.
[10] S. Sawada, S. Settepanella and S. Yamagata, Discriminantal arrangement, 3 x 3 minors of Plücker matrix and hypersurfaces in Grassmannian Gr(3, n), C. R. Acad. Sci. Paris Ser. 1355 (2017), 1111-1120, doi:10.1016/j.crma.2017.10.011.