Lines in Higgledy-piggledy Arrangement

In this article, we examine sets of lines in PG(d, F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least 1.5d lines if the field F has at least 1.5d elements, and at least 2d − 1 lines if the field F is algebraically closed. We show that suitable 2d − 1 lines constitute such a set (if |F| 2d − 1), proving that the lower bound is tight over algebraically closed fields. At last, we will see that the strong (s, A) subspace designs constructed by Guruswami and Kopparty have better (smaller) parameter A than one would think at first sight.


Introduction
Héger, Patkós and Takáts [1] hunt for a set G of points in the projective space PG(d, q) that 'determines' all hyperplanes in the sense that the intersection Π ∩ G is individual for each hyperplane Π.
A little different but similar problem is to find a set G such that each hyperplane is spanned by the intersection Π ∩ G.Such a 'generator set' is always a 'determining set' since if all the intersections Π ∩ G span the hyperplanes Π then they must be individual.Héger, Patkós and Takáts thus began to examine 'generator sets'.In projective planes generator sets and two-fold blocking sets are the same, since two distinct points span the line connecting these points.
Definition 1 (Multiple blocking set).A set B of points in the projective space P is a t-fold blocking set with respect to hyperplanes, if each hyperplane Π ⊂ P meets B in at least t points.One can define t-fold blocking sets with respect to lines, planes, etc. similarly.
The definition of the t-fold blocking set does not say anything more about the intersections with hyperplanes.In a projective space of dimension d 3, a d-fold blocking set can intersect a hyperplane Π in such a set of d points which is contained in a proper subspace of Π.Thus (in higher dimensions), a natural specialization of multiple blocking sets would be the following.(Since in higher dimension a projective space is always over a field, we use the special notation PG(d, F) instead of the general P.) Definition 2 (Generator set).A set G of points in the projective space PG(d, F) is a generator set with respect to hyperplanes, if each hyperplane Π ⊂ PG(d, F) meets G in a 'generator system' of Π, that is, G ∩ Π spans Π, in other words this intersection is not contained in any hyperplane of Π. (Hyperplanes of hyperplanes are subspaces in PG(d, F) of co-dimension two.)Example 3. In a projective plane PG(2, q 2 ) there exist two disjoint Baer-subgeometries.These together constitute a 2-fold blocking set, and thus, a generator set consisting of 2q 2 + 2q + 2 points.
Remark 4. In PG(d, q d ), d disjoint subgeometries of order q together constitute a d-fold blocking set.But it is not obvious whether this example is only a d-fold blocking set or it could be also a generator set (if we choose the subgeometries in a proper way).
Héger, Patkós and Takáts [1] had the idea to search for generator set which is the union of some disjoint lines and they gave an example for such a 'determining set' as the union of the points of 2d + 2 distinct lines, using probabilistic method.They gave the name 'higgledy-piggledy' to the property of such sets of lines.We investigate their idea.

Hyperplane-generating sets of lines
The trivial examples for multiple blocking sets are the sets of disjoint lines: If B is the set of points of t disjoint lines then B is a t-fold blocking set (with respect to hyperplanes).Héger, Patkós and Takáts [1] suggested to search generator sets in such a form.(Though there can exist smaller examples.) Sets of k disjoint lines are always multiple (k-fold) blocking sets (with respect to hyperplanes) but not always generator sets, so the following definition is not meaningless.Definition 5 (Generator set of lines).A set L of lines is a generator set (with respect to hyperplanes), if the set L of all points of the lines contained by L is a generator set with respect to hyperplanes.Elements of a generator set of lines are said to be in higgledy-piggledy arrangement.
From now on, we will examine sets of lines of the property above.

Examples in projective planes
Let P be an arbitrary (desarguesian or not, finite or infinite) projective plane and let 1 and 2 be two distinct lines and let Q = 1 ∩ 2 denote the meeting point.Each line of P not containing Q meets 1 and 2 in two distinct points, thus, is generated.Lines containing Q meet 1 and 2 only in Q, so they are not generated.This shows that two lines cannot be in higgledy-piggledy arrangement.
Example 6 (Triangle).Let 3 be an arbitrary line not containing Q.Other lines containig Q meet 3 , thus, they are also generated by { 1 , 2 , 3 }.Thus, three lines in general position constitute a generator set in arbitrary projective plane.
Remark 7. If P has only three lines through a point (i.e.P is the Fano plane), three concurrent lines also form a generator set.
In the projective plane PG(2, q), a minimal generator set of lines contains three lines and thus 3q points (if these three lines have three distinct meeting points).Whereas two disjoint Baer subplanes (containing only 2q + 2 √ q + 2 points) together also constitute a generator set (of points) with respect to lines.This example shows that there can exist generator set (of points) with respect to hyperplanes, containing less points than the smallest generator set of lines.

Examples in projective spaces of dimension three
Let 1 , 2 , 3 be pairwise disjoint lines in PG(3, F), and let Q + (3, F) be the (unique) hyperbolic quadric containing these lines.Each plane of PG(3, F) which is not a tangent plane of Q + (3, F) meets these three lines in non-collinear three points, thus it is generated.Let denote one of the opposite lines meeting 1 , 2 and 3 .Planes through containing neither 1 , nor 2 , nor 3 meet these lines in collinear points (on the opposite line ), and thus, they are not generated.
Remark 8.The reader can show that if these three lines are not pairwise disjoint, they cannot constitute a generator set: See the planes through the meeting point of two lines.
Example 9 (Over F q and over R or Q).If there exists a line 4 disjoint to the hyperbolic quadric Q + (3, F), then each plane Π not generated by { 1 , 2 , 3 } (meeting them in three collinear points) meet 4 in a point Q 4 not on the line of the three collinear meeting points the electronic journal of combinatorics 21(2) (2014), #P2.56 The example above does not exist if the field F is algebraically closed since in this case the hyperbolic quadric Q + (3, F) meets every lines.

Lower bound over arbitrary (large enough) fields
At first, we try to give another equivalent definition to the higgledy-piggledy property of generator sets of lines.The following is not an equivalent but a sufficient condition.Although, in several cases it is also a necessary condition (if we seek minimal sets of this type), thus, it could effectively be considered as an almost-equivalent condition.
Theorem 11 (Sufficient condition).If there is no subspace of co-dimension two meeting each element of the set L of lines then L is a generator set with respect to hyperplanes.
Proof.Suppose that the set L of lines is not a generator set with respect to hyperplanes.Then there exists at least one hyperplane Π that meets the elements of L in a set Π ∩ L of points which is contained in a hyperplane H of Π.Since Π is a hyperplane it meets every line, thus each element of L meets Π, but the point(s) of intersection has (have) to be contained in H. Thus the subspace H (of co-dimension two) meets each element of L.
The theorem above is a sufficient but not necessary condition.But if this condition above does not hold, then the set L of lines could only be generator set in a very special way.
Lemma 12.If the set L of lines is a generator set with respect to hyperplanes and there exists a subspace H of co-dimension two that meets each element of L then L has to contain at least as many lines as there are points in a projective line.(That is, |L| q + 1 if the field F = F q and L is infinite if the field F is not finite.) Proof.Let be a line not intersecting H.For each point P i ∈ there exists a hyperplane Π i containing H and meeting P i .For each such hyperplane Π i there exists a line i ∈ L that meets Π i not only in H, thus i ⊂ Π i .Two distinct hyperplanes Π i and Π j intersect in H thus the lines i and j have to be different lines.
If we seek minimal size generator sets (and the field F has at least 1.5d elements where d is the dimension) we can suppose the condition of Theorem 11, so we seek minimal size set of lines such that no subspace of co-dimension two meets each line.(if these lines are contained in a less dimensional subspace, it can be extended).If d is even, this subspace is a hyperplane Π.If d is odd, this subspace has co-dimension two, and thus it can be extended to a hyperplane Π.The hyperplane Π meets each line, thus let for i = 1, . . ., d − 1.There exists a hyperplane H of Π that contains each point P i above.(If these points would be not in general position, that is not a problem.)The subspace H has co-dimension two in PG(d, F) and it meets the lines 1 , . . .The examples in PG(2, q) and PG (3, q) show that this lower bound is tight in small dimensions (d 3) over finite fields, and over R and over Q.
Remark 15.As in PG(2, 2) three lines through a point are also in higgledy-piggledy arrangement, four proper lines having a common transversal line meeting them can be in higgledy-piggledy arrangement in PG(3, 3).

Grassmann varieties
The sufficient condition is an intersection-property of some subspaces.Such properties can naturally be handled using Grassmann varieties and Plücker co-ordinates.The original (hyperplane generating) property can also be translated to the language of Plücker coordinates.
Let G(m, n, F) or simply G(m, n) denote the Grassmannian of the linear subspaces of dimension m and co-dimension n in the vector space F m+n , or, in other aspect G(m, n) is the set of all projective subspaces of dimension m − 1 (and co-dimension n) in PG(m + n − 1, F). Via 'Plücker embedding' we can identify this Grassmannian to the set of one dimensional linear subspaces of m F m+n generated by totally decomposable multivectors, that is, The canonical isomorphism m F m+n ≡ n F m+n defines a bijection between G(m, n) and G(n, m).Thus, the Grassmannian of subspaces of co-dimension two can be considered as the Grassmannian of the lines of the dual projective space.
together with the quadratic Plücker relations (for each quadruple i 1 i 2 i 3 i 4 of indices) Proof.According to [2, Theorem 3.1.6.], the Plücker relations completely determine the Grassmannian (moreover, they generate the ideal of polynomials vanishing on it).In case n = 2, the Plücker relations found in [2, Subsection 3.1.3.]reduces to the form Finally, i<j
Remark 18.One can see that in suitable positions the Plücker co-ordinate vector L(t) has the co-ordinates: 1, t 2 , t 4 , t 6 , . . ., t 2d−2 and the co-ordinates: 2t, 2t 3 , . . ., Lemma 19.If either char F = p > d and |F| 2d − 1 or char F = 0, then there does not exist any subspace of co-dimension two meeting each tangent line t of the moment curve.
Notice that in equations ( 1), ( 2 Using these equations and the Plücker relatios, we can prove by induction, that all Plücker co-ordinates H ij are zero, and thus, they are not the homogeneous co-ordinates of any subspace H.We do two inductions, one for N = 1, . . ., d (increasing) and another (decreasing) one for N = (2d − N ) = 2d − 1, . . ., d + 1. Remember that char F = 0 or char F > d, so the nonzero integers in these equations are nonzero elements of the prime field of F.

Increasing induction
The first two equations say that H 01 = H 02 = 0. Suppose by induction that we have H ij = 0 for each pair (i, j) where 0 i < j N − i, where N is a positive integer less than d.Using this assumption, we prove that H 0,N +1 = H 1,N = H 2,N −1 = • • • = 0, and thus H ij = 0 for each pair (i, j) where 0 i < j N + 1 − i.
Equation (N + 1) says that a linear combination of the Plücker co-ordinates H 0,N +1 , H 1,N , H 2,N −1 , . . ., H N +1 2 , N +1 2 is zero.Let H ij and H kl be two arbitrary element among these above.We have the Plücker relation Thus, these Plücker relations say that all H ij (among H 0,N +1 , H 1,N , . . ., 2 ) should be zero except one.And the linear Equation (N + 1) says that this one cannot be exception either.

Decreasing induction
The decreasing induction, started with the last two equations So we have proved that each Plücker co-ordinate of the subspace H of co-dimension two should be zero, that is a contradiction, since Plücker co-ordinates are homogeneous.
Theorem 20.If either char F = p > d and |F| 2d−1 or char F = 0, then arbitrary 2d−1 distinct tangent lines t together constitute a generator set with respect to hyperplanes.Proof.Let { t i : i = 1, 2, . . ., 2d − 1} be an arbitrary set of 2d − 1 tangent lines of the rational normal curve.It is enough to prove that there is no subspace H of co-dimension two meeting each element of this set.
Suppose to the contrary that there exists such a subspace H and let H ij be the Plücker co-ordinates of it.Since H meets each line t i , this means i<j H ij L ij (t k ) = 0 for all t k , k = 1, . . ., 2d − 1.Thus, the polynomial d−1 i=0 d j=i+1 H ij (j − i)t i+j−1 has 2d − 1 roots, but its degree is at most 2d − 2. So, if there exists such a subspace H of co-dimension two, the polynomial above is the zero polynomial, and thus, H meets each tangent line t , contradicting Lemma 19.
These results above require the characteristic char F to be greater than the dimension d (or to be zero).However, we can generalize these results over small prime characteristics.

Small prime characteristics: 'diverted tangents'
The only weakness of the proof of Lemma 19 (which can be ruined by small prime characteristic) is the linear equation system for the Plücker co-ordinates H ij .The Plücker co-ordinate H ij has coefficient j − i mod p and this could be zero for j = i if the characteristic p is not greater than the dimension d.
Remark 21.If the characteristic of F equals to the dimension d, then there exists exactly one subspace of co-dimension two that meets each tangent t of the moment curve.The Plücker co-ordinates of this subspace should be all zero except one: H 0,d .This subspace H thus can be get as the intersection of two hyperplanes co-ordinatized by [1, 0, . . ., 0] (the ideal hyperplane) and [0, . . ., 0, 1].
In higher dimension there will be more such subspaces, and thus, their intersection is a subspace of codimension more than two, meeting each tangent line.
If we substitute the coefficients (j − i) by nonzero elements, the proof of Lemma 19 will be valid over arbitrary characteristic.Remember that the Plücker co-ordinates of the tangent line t are L ij (t) = (j − i)t i+j−1 and the coefficient (j − i) comes from here.
Definition 22 (Diverted tangent lines).Consider the line t connecting a(t) and b(t) instead of the tangent line t of the moment curve in the point a(t).The Plücker coordinate vector of the 'diverted tangent line' Diverted tangent lines depend on the injection ϕ.
the electronic journal of combinatorics 21(2) (2014), #P2.56 Remark 23.If char F is zero, the injection ϕ can be the identity, and if char F = p > d, the injection ϕ can be defined by ϕ(k) ≡ k mod p.In these cases the diverted tangent line t determined by ϕ equals to the actual tangent line t of the moment curve.
Proof.Suppose to the contrary that the subspace H meets the diverted tangent lines , but its degree is at most 2d − 2. So, the polynomial above is the zero polynomial, and thus, H meets each connecting line t Now, we can repeat the proof of Lemma 19 by substituting the coefficients (j − i) by ϕ(j) − ϕ(i) in the linear equations ( 1), (2), . . ., (2d − 1), and since ϕ is injective, these coefficients are nonzero.Thus, we can prove that each Plücker co-ordinate H ij should be zero, which is a contradiction.
We have proved that over arbitrary (large enough) field we can construct a hyperplanegenerating set of lines of size 2d−1.In the next section, we will prove that it is the smallest one if the field is algebraically closed.

Lower bound over algebraically closed fields
A projective line has infinitely many points over an algebraically closed field, so Lemma 12 concludes that (over algebraically closed field) the finite set L of lines could be a generator set only if the sufficient (almost-equivalent) condition of Theorem 11 holds.Remember that an algebraic surface G ⊂ P of dimension n and a projective subspace S P of co-dimension n always meet over an algebraically closed field.
Theorem 26.Over algebraically closed field F, if the set L of lines in PG(d, F) has at most 2d − 2 elements, then there exists a subspace H in PG(d, F) of co-dimension two that meets each element of L, and thus, L is not a generator set.
Remark 31.If we have a weak (s, A) subspace-design of M subspaces (M > A), then any A + 1 subspaces among them constitute a weak (s, A) subspace design.Thus, if we have a weak (2, N −1) subspace design of M N subspaces of co-dimension two, we will also have a set of N lines in higgledy-piggledy arrangement.
We are interested in (2, N −1) subspace designs containing subspaces of co-dimension two, thus s = 2 = rt, and thus r = 1 and t = 2.In this case the Guruswami-Kopparty Theorem 30 gives a strong (2, 2d) subspace design containing M > const • q subspaces of co-dimension two.If M > 2d, this design (after dualization) gives us a set of 2d + 1 lines in higgledy-piggledy arrangement.
Watching the Guruswami-Kopparty constructions [3, Sections 4-5] with both eyes, we can behold the fact that these constructions yield a little bit stronger version of Theorem 30.This will be shown in the following two subsections.

Constructions based on Folded Reed-Solomon codes
Guruswami and Kopparty based their main result [3,Theorem 7] on the following construction presented in [3,Section 4].We will use d instead of m − 1.Let s t d + 1 < q and r be positive integer parameters and identify F d+1 q with the F q -linear subspace of polynomials of degree d in F q [X] and let ω denote a generator of F * q .For α ∈ F q r , let S α ⊆ F q r be given by S α = {α q j ω i | 0 j < r, 0 i < t}.
Let F ⊆ F q r be a large set such that: • For each α ∈ F: F q (α) = F q r .
• Each S α has cardinality rt.
For each α ∈ F let subspace design.
We do not repeat the proof here, for details see [3, pages 8-10].The keystone of the proof of this theorem above is the following matrix.Let W F d+1 q be a subspace and let the polynomials P 1 , . . ., P s constitute a basis of W . Define the following t × s matrix of polynomials: Let A(X) be the top s × s submatrix of M (X) and let L(X) be the determinant of A(X).
The term d • s in the parameter s, d•s r(t−s+1) comes directly from the fact that the polynomial L(X) has degree at most d • s.We can give a better bound for this degree: Lemma 33.The polynomial L(X) has degree at most ds − s 2 .Proof.The basis P 1 , . . ., P s of the subspace W F d+1 q can be chosen (by Gaussian elimination) such that deg(P 2 .As a consequence, the Guruswami-Kopparty Theorem 32 above will have the following improved form. Corollary 34 (Guruswami-Kopparty; improved version).Using the notation above, the subspace design.
This observation shows that the Guruswami-Kopparty construction of [3, Section 4] based on Folded Reed-Solomon codes actually give us a strong (2, 2d−1) subspace design, and thus, a set of 2d lines in higgledy-piggledy arrangement.

Constructions based on Multiplicity codes
The main result of [3] is also proved by the following construction presented in [3, Section 5] which could be used only over large characteristics.We will again use d instead of m − 1.Let 0 < s t d + 1 < char F q be integer parameters and identify F d+1 q with the F q -linear subspace of polynomials of degree d in F q [X].For each α ∈ F q let H α = {P (X) ∈ F d+1 q | mult(P, α) t} Theorem 35 (Guruswami-Kopparty [3, Theorem 17]).For every F q -linear subspace W F d+1 q with dim(W ) = s we have We do not repeat the proof here, for details see [3, pages 11-12].The proof of this theorem uses the the following matrix.Let W F d+1 q be a subspace and let the polynomials P 1 , . . ., P s constitute a basis of W . Define the following t × s matrix of polynomials: . . .P s (X) P 1 (X) . . .P s (X) . . . . . .As a consequence, the Guruswami-Kopparty Theorem 35 above will have the following improved form.
Corollary 37 (Guruswami-Kopparty; improved).For every F q -linear subspace W F d+1 q with dim(W ) = s we have Theorem 38 (Guruswami-Kopparty; improved).For all positive integers s, r, t, m = d+1 and prime powers q satisfying s t m < q, there is an explicit collection of M = Ω q r rt linear subspaces H 1 , . . ., H M ⊂ F m q , each of co-dimension rt, which forms a strong (s, A) subspace design, where A (m−1− s−1 2 )s r(t−s+1) , and even A (m−s)s r(t−s+1) if m < char F q .
So, we have shown that over large enough characteristic, the construction of [3, Section 5] based on multiplicity codes actually give us a strong (2, 2d − 2) subspace design, and thus, a set of 2d − 1 lines in higgledy-piggledy arrangement.

Open questions
As we have seen previously, subspace designs constructed by Guruswami and Kopparty [3] can also give us hyperplane-generating set of lines of size 2d − 1 (if char F > d + 1), the optimal size over algebraically closed field.But examples in low dimensions show that much smaller hyperplane-generating sets of lines could exist, if the field is finite.

Lemma 13 . 2 and d 2 +i ( 1 i d − 1 )
If the set L of lines in PG(d, F) has at most d 2 + d − 1 elements then there exists a subspace H of co-dimension two meeting each line in L. the electronic journal of combinatorics 21(2) (2014), #P2.56 Proof.Let 1 , . . ., d denote the elements of L. There exists a subspace of dimension 2 d 2 − 1 containing the lines 1 , . . ., d 2

, d 2
since these lines are contained in Π and H is a hyperplane of Π, and H meets the other lines since the meeting points are the points P i .Theorem 14 (Lower bound).If the field F has at least d 2 +d elements, then a generator set L of lines in PG(d, F) has to contain at least d 2 + d elements.Proof.Lemma 12 and Lemma 13 together give the result.

Remark 16 .
If m = 2 or n = 2 then the Plücker co-ordinate vectors can be considered as alternating matrices: L ij = a i b j − a j b i where L = a ∧ b.Proposition 17.Let {L(1), . . ., L(k)} denote the set of the Plücker co-ordinate vectors representing the elements of the set L of k lines in PG(d, F).There exists a subspace H of co-dimension two in PG(d, F) meeting each element of L if and only if the subspace L indices.Since we consider the Grassmannian G(d−1, 2) of subspaces of co-dimension two as the Grassmannian G(2, d−1) of lines of the dual space, the Plücker relations determining G(d−1, 2) are the same (using dual co-ordinates).Let a, b ∈ F d+1 be the homogeneous co-ordinate vectors of two projective points in PG(d, F) and let x, y ∈ F d+1 be the homogeneous (dual) co-ordinate vectors of two hyperplanes in PG(d, F).The line connecting P(a) and P(b) is defined by the Plücker co-ordinate vector a ∧ b ∈ G(2, d − 1).The subspace of co-dimension two defined by the Plücker coordinate vector x ∧ y ∈ G(d − 1, 2) is the intersection of the hyperplanes x ⊥ and y ⊥ .The line co-ordinatized by L = a ∧ b and the subspace co-ordinatized by H = x ∧ y meet each other if and only if the scalar product x ∧ y|a ∧ b = x|a y|b − x|b y|a equals to zero, because there exists a point P(αa + βb) ∈ L contained also in H if and only if x|αa + βb = α x|a + β x|b = 0 and y|αa + βb = α y|a + β y|b = 0 for a suitable pair (α, β) = (0, 0).

2 .
X) be the top s × s submatrix of M (X) and let L(X) be the determinant of A(X). the electronic journal of combinatorics 21(2) (2014), #P2.56The term d • s in the parameter d•s t−s+1 in Theorem 35 above comes from the fact deg(L(X)) ds.As in the previous subsection, there is again a better bound for this degree: Lemma 36.The polynomial L(X) has degree at most s(d − s + 1).Proof.Expanding the determinant L(X) = π∈Ss (−1) ) (X) has degree s k=1 deg(P π(k) ) − (k − 1) , that is equal to s k=1 deg(P k ) − s The basis P 1 , . . ., P s of the subspace W F d+1 q can be chosen (by Gaussian elimination) such that deg(P 1 ) < deg(P 2 ) < • • • < deg(P s ) d and thus, deg(L) i deg(P i ) − s 2 d + • • • + d − (s − 1) d − s + 1).