Classification of k-nets

A finite \emph{$k$-net} of order $n$ is an incidence structure consisting of $k\ge 3$ pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the $k$ classes. Deleting a line class from a $k$-net, with $k\ge 4$, gives a \emph{derived} ($k-1$)-net of the same order. Finite $k$-nets embedded in a projective plane $PG(2,K)$ coordinatized by a field $K$ of characteristic $0$ only exist for $k=3,4$, see \cite{knp_k}. In this paper, we investigate $3$-nets embedded in $PG(2,K)$ whose line classes are in perspective position with an axis $r$, that is, every point on the line $r$ incident with a line of the net is incident with exactly one line from each class. The problem of determining all such $3$-nets remains open whereas we obtain a complete classification for those coordinatizable by a group. As a corollary, the (unique) $4$-net of order $3$ embedded in $PG(2,K)$ turns out to be the only $4$-net embedded in $PG(2,K)$ with a derived $3$-net which can be coordinatized by a group. Our results hold true in positive characteristic under the hypothesis that the order of the $k$-net considered is smaller than the characteristic of $K$.


Introduction
Finite 3-nets occur naturally in combinatorics since they are geometric representations of important objects as latin squares, quasigroups, loops and strictly transitive permutation sets.Historically, the concept of 3-net arose from classical differential geometry via the combinatorial abstraction of the concept of a 3-web; see [14].In recent years finite 3-nets embedded in a projective plane PG(2, K) coordinatized by a field K were investigated in algebraic geometry and resonance theory, see [5,12,15,18,19], and a few infinite families of such 3-nets were constructed and classified, see [10].
In this paper, we deal with finite 3-nets embedded in PG(2, K) such that the three line classes of the 3-net appear to be in perspective position with axis r, that is, whenever a point P ∈ r lies on a line of the 3-net then P lies on exactly one line from each line classes of the 3-net.If a 3-net is in perspective position then the corresponding latin square has a transversal, equivalently, at least one of the quasigroups which have the latin square as a multiplicative table has a complete mapping; see [2,Section 1.4].A group has a complete mapping if and only its 2-subgroups of Sylow are either trivial or not cyclic.This was conjectured in the 1950's by Hall and Paige [6], see [2, p. 37], and proven only recently by Evans [4].
As in [10], most of the known examples in this paper arise naturally in the dual plane of PG(2, K), and it is convenient work with the dual concept of a 3-net embedded in PG(2, K).Formally, a dual 3-net in PG(2, K) consists of a triple (Λ 1 , Λ 2 , Λ 3 ) with Λ 1 , Λ 2 , Λ 3 pairwise disjoint point-sets, called components, such that every line meeting two distinct components meets each component in precisely one point.Every component has the same size n, the order of the dual 3-net, and each of the n 2 lines meeting all components is a line of the dual 3-net.
A dual 3-net (Λ 1 , Λ 2 , Λ 3 ) is in perspective position with a center C, where C is a point off Λ 1 ∪ Λ 2 ∪ Λ 3 , if every line through C meeting a component is a line of the dual 3-net, that is, still meets each component in exactly one point.A dual 3-net in perspective position has a transversal.Furthermore, a dual 3-net may be in perspective position with different centers although the number of such centers is bounded by the order of the 3-net.If this bound is attained and every lines through two centers is disjoint from Λ 1 ∪ Λ 2 ∪ Λ 3 , then the set of the centers can be viewed as a new component Λ 4 to add to (Λ 1 , Λ 2 , Λ 3 ) so that the resulting quadruple (Λ 1 , Λ 2 , Λ 3 , Λ 4 ) is a dual 4-net, that is, a 4-net in the dual plane.
From previous work [11,Proposition 3.1], finite dual 4-nets have constant cross-ratio, that is, for every line ℓ intersecting the components, the crossratio (P 1 , P 2 , P 3 , P 4 ) with P i = Λ i ∩ ℓ is the same.For a dual 3-net in perspective position with a center C, this raises the problem whether for all lines ℓ through C, the cross-ratio of the points C, ℓ ∩ Λ 1 , ℓ ∩ Λ 2 , ℓ ∩ Λ 3 is the same.By our Proposition 2.3, the answer is affirmative.Moreover, in case of more than one centers, the cross-ratio does not depend on which center is referred to.Therefore, any dual 3-net in perspective position has constant cross-ratio.
The problem of classifying all 3-nets in perspective position remains open and appears to be difficult.Its solution would indeed imply the answer to the main conjecture on finite 4-nets, namely the non-existence of 4-nets of order greater than three.Our main result in this context is the following theorem that provides a complete classification for those 3-nets in perspective position which are coordinatizable by a group.Theorem 1.1.Let Λ be a dual 3-net of order n which is coordinatized by a group.Assume that Λ is embedded in a projective plane PG(2, K) over an algebraically closed field whose characteristic is either 0 or bigger than n.If Λ is in perspective position and n = 8 then one of the following two cases occur: (i) A component of Λ lies on a line while the other two lie on a nonsingular conic.More precisely, Λ is projectively equivalent to the dual 3-net given in Lemma 4.2.
(ii) Λ is contained in a nonsingular cubic curve C with zero j(C)-invariant, and Λ is in perspective position with at most three centers.
Theorem 1.1 provides evidence on the above mentioned conjecture about 4-nets.In fact, it shows for n = 8 that the (unique) 4-net of order 3 embedded in PG(2, K) is the only 4-net embedded in PG(2, K) which has a derived 3-net coordinatized by a group G.This result remains valid in positive characteristic under the hypothesis that that the order n of the k-net considered is smaller than the characteristic of K, apart from possibile sporadic cases occurring for n ∈ {12, 24, 60} and G ∼ = Alt 4 , Sym 4 , Alt 5 , respectively.
The proof of Theorem 1.1 follows from Propositions 3.2, 3.3, 4.3 and Theorem 5.3 together with the classification of 3-nets coordinatized by groups, see [10] and [13], which states that the dual of such a 3-net is either algebraic (that is, contained in a reducible or irreducible cubic curve), or of tetrahedron type, or n = 8 and G is the quaternion group of order 8.This classification holds true in positive characteristic if the characteristic of K exceeds the order n of 3-net and none of the above mentioned special cases for n = 12, 24, 60 occurs.

The constant cross-ratio property
In [11,Proposition 3.1], the authors showed that (dual) 4-nets have constant cross-ratio, that is, for any line intersecting the components, the crossratio of the four intersection points is constant.In this section we prove a similar result for (dual) 3-nets in perspective positions.Our proof relies on some ideas coming from [11].
Proposition 2.1.Let F, G be homogeneous polynomials of degree n such that the curves F : F = 0 and G : G = 0 have n 2 different points in common.Fix nonzero scalars α, β, α ′ , β ′ ∈ K, and define the polynomials and the corresponding curves H : H = 0, H ′ : H ′ = 0.Then, for all P ∈ F ∩ G, the tangent lines t P (F ), t P (G), t P (H), t P (H ′ ) have cross-ratio Proof.We start with three observations.Notice first that for any P ∈ F ∩ G, the intersection multiplicity of F and G at P must be 1 by Bézout's theorem.This implies that P is a smooth point of both curves, and that the tangent lines t P (F ), t P (G) are different.Second, the polynomials F, G, H, H ′ are defined up to a scalar multiple.Multiplying them by scalars such that the curves F , G, H, H ′ don't change, the value of the cross-ratio κ remains invariant as well.And third, the change of the projective coordinate system leaves the homogeneous pairs (α, β), (α ′ , β ′ ) invariant, hence it does not affect κ.
Let us now fix an arbitrary point P ∈ F ∩ G and choose the projective coordinate system such that P = (0, 0, 1), t P (F ) : X = 0, t P (G) : Y = 0. We set Z = 0 as the line at infinity and switch to affine coordinates x = X/Z, y = Y /Z.For the polynomials we have with polynomials f 2 , g 2 , h 2 , h ′ 2 of lower degree at least 2. This shows that the respective tangent lines have equations hence, the cross-ratio is indeed κ.
Let ℓ be a transversal line of (λ 1 , λ 2 , λ 3 ), that is, assume that ℓ intersects all lines of the 3-net in the total of n points P 1 , . . ., P n .Let Q be another point of ℓ, that is, Q = P i , i = 1, . . ., n.There are unique scalars α ′ , β ′ such that the curve Moreover, since H 0 cannot pass through P 1 , . . ., P n , the tangent lines of H ′ at these points are equal to ℓ. Proposition 2.1 implies the following.Proposition 2.2 (Constant cross-ratio for 3-nets with transversal).Let λ = (λ 1 , λ 2 , λ 3 ) be a 3-net of order n, embedded in PG(2, K).Assume that ℓ is a transversal to λ.Then there is a scalar κ such that for all The dual formulation of the above result is the following Proposition 2.3 (Constant cross-ratio for dual 3-nets in perspective position).Let Λ = (Λ 1 , Λ 2 , Λ 3 ) be a dual 3-net of order n, embedded in PG(2, K).Assume that Λ is in perspective position with respect to the point T .Then there is a scalar κ such that for all lines ℓ through T , the cross-ratio of the points In the case when a component of a dual 3-net is contained in a line, the constant cross-ratio property implies a high level of symmetry of the dual 3-net.The idea comes from the argument of the proof of [11,Theorem 5.4].
Assume that Λ is in perspective position with respect to the point T , and, Λ 1 is contained in a line ℓ.Then there is a perspectivity u with center T and axis ℓ such that Λ u 2 = Λ 3 .
Proof.Let κ be the constant cross-ratio of Λ w.r.t T .Define u as the (T, ℓ)perspectivity which maps the point P to P ′ such that the cross-ratio of the points T , P , P ′ and T P ∩ ℓ is κ.Then Λ u 2 = Λ 3 holds.

Triangular and tetrahedron type dual 3-nets in perspective positions
With the terminology used in [10], a dual 3-net is regular if each of its three components is linear, that is, contained in a line.Also, a regular dual 3-net is triangular or of pencil type according as the lines containing the components form a triangle or are concurrent.Proposition 3.1.Any regular dual 3-net in perspective position is of pencil type.
Proof.Let (Λ 1 , Λ 2 , Λ 3 ) be a dual 3-net in perspective position with center P such that Λ i is contained in a line There exists a perspectivity ϕ with center R and axis r through P which takes L 1 to L 2 and M 1 to M 2 .For i = 1, 2, the line t i through P, L i , M i meets Λ 3 in a point N i .From Proposition 2.3 the cross ratios (P L 1 M 1 N 1 ) and (P L 2 M 2 N 2 ) coincide.Therefore, ϕ takes N 1 to N 2 and hence the line From [10, Lemma 3], dual 3-nets of pencil type do not exist in zero characteristic whereas in positive characteristic they only exist when the order of the dual 3-net is divisible by the characteristic.This, together with Proposition 3.1, give the following result.Proposition 3.2.No regular dual 3-net in perspective position exists in zero characteristic.This holds true for dual 3-nets in positive characteristic whenever the order of the 3-net is smaller than the characteristic.Now, let Λ = (Λ 1 , Λ 2 , Λ 3 ) be tetrahedron type dual 3-net of order n ≥ 4 embedded in PG(2, K).From [10, Section 4.4], n is even, say n = 2m, and is a dual 3-net of triangular type.An easy counting argument shows that each of the n 2 lines of Λ is a line of (exactly) one of the dual 3-nets Φ i with 1 ≤ i ≤ 4. Now, assume that Λ is in perspective position with a center P .Then each of the n lines of Λ passing through P is also a line of (exactly) one Φ i with 1 ≤ i ≤ 4. Since n > 4, some of these triangular dual 3-nets, say Φ, has at least two lines through T .Let ℓ 1 , ℓ 2 , ℓ 3 denote the lines containing the components of Φ, respectively.Now, the proof of Proposition 3.1 remains valid for Φ showing that Φ is of pencil type.But this is impossible as we have pointed out after that proof.Therefore, the following result is proven.Proposition 3.3.No tetrahedron type dual 3-net in perspective position exists in zero characteristic.This holds true for dual 3-nets in positive characteristic whenever the order of the 3-net is smaller than the characteristic.Propositions 3.2 and 3.3 have the following corollary.
Corollary 3.4.Let K be an algebraically closed field whose characteristic is either zero or greater than n.Then no dual 4-net of order n embedded in PG(2, K) has a derived dual 3-net which is either triangular or of tetrahedron type.

Conic-line type dual 3-nets in perspective positions
Let Λ = (Λ 1 , Λ 2 , Λ 3 ) be dual 3-net of order n ≥ 5 in perspective position with center T , and assume that Λ is of conic-line type.Then Λ has a component, say Λ 1 , contained in a line ℓ, while the other two components Λ 2 , Λ 3 lie on a nonsingular conic C. From Proposition 2.3, Λ has constant cross-ratio κ with respect to T .Proof.By Proposition 2.4, there is a (T, ℓ)-perspectivity u which takes Λ 2 to Λ 3 .The image C ′ of the conic C by u contains Λ 3 and hence Λ 3 is in the intersection of C and C ′ .Since Λ 3 has size n ≥ 5, these nonsingular conics must coincide.Therefore, u preserves C and its center T is the pole of its axis ℓ is in the polarity arising from C. In particular, u is involution and hence κ = −1.
Choose our projective coordinate system such that T = (0, 0, 1), ℓ : Z = 0 and that C has equation XY = Z 2 .In the affine frame of reference, ℓ is the line at infinity, T is the origin and C is the hyperbola of equation xy = 1.Doing so, the map (x, y) → (−x, −y) is the involutorial perspectivity u introduced in the proof of Proposition 4.1.
As it is shown in [10, Section 4.3], Λ has a parametrization in terms of an n th -root of unity provided that the characteristic of K is either zero or it exceeds n.In our setting Λ 2 = {(c, c −1 ), (cξ, c −1 ξ −1 ), . . ., (cξ n−1 , c −1 ξ −n+1 )}, where c ∈ K * and ξ is an nth root of unity in K.Moreover, since u takes Λ 2 to Λ 3 , It should be noted that such sets Λ 2 and Λ 3 may coincide, and this occurs if and only if n is even since ξ n/2 = −1 for n even.Therefore n is odd and then where (m) denotes the infinite point of the affine line with slope m.Therefore the following results are obtained.Lemma 4.2.For n odd, the above conic-line type dual 3-net is in perspective position with center T .Proposition 4.3.Let K be an algebraically closed field of characteristic zero or greater than n.Then every conic-line type dual 3-net of order n which is embedded in PG(2, K) in perspective position is projectively equivalent to the example given in Lemma 4.2.In particular, n is odd.Proof.Since the only 4-net of order 3 contains no linear component, and there exist no 4-nets of order 4, we may assume that n ≥ 5. Let Λ 1 , Λ 2 , Λ 3 , Λ 4 be a dual 4-net with a derived dual 3-net of conic-line type.Without loss of generality, Λ 1 lies on a line.Then the derived dual 3-net (Λ 1 , Λ 2 , Λ 3 ) is of conic-line type.From Proposition 4.3, there exists a non-singular conic C containing Λ 2 ∪ Λ 3 .Similarly, (Λ 1 , Λ 2 , Λ 4 ) is of conic-line type and Λ 2 ∪ Λ 4 is contained in a non-singular conic D. Since C and D share Λ 2 , the hypothesis n ≥ 5 yields that C = D. Therefore, the dual 3-net (Λ 2 , Λ 3 , Λ 4 ) is contained in C.But this contradicts [1, Theorem 6.1].
It is well known that plane cubic curves have quite a different behavior over fields of characteristic 2, 3. Since the relevant case in our work is in zero characteristic or in positive characteristic greater than the order of the 3-net considered, we assume that the characteristic of K is neither 2 nor 3. From classical results, Γ is either nonsingular or it has at most one singular point, and in the latter case the point is either a node or a cusp.Accordingly, the number of inflection points of Γ is 9, 3 or 1.If Γ has a cusp then it has an affine equation Y 2 = X 3 up to a change of the reference frame.Otherwise, Γ may be taken in its Legendre form so that Γ has an infinite point in Y ∞ = (0, 1, 0) and it is nonsingular if and only if c(c − 1) = 0.The projective equivalence class of a nonsingular cubic is uniquely determined by the j-invariant; see [7,Section IV.4].Recall that j arises from the cross-ratio of the four tangents which can be drawn to Γ from a point of Γ and it takes into account the fact that four lines have six different permutations.Formally, let t 1 , t 2 , t 3 , t 4 be indeterminates over K.The cross-ratio of t 1 , t 2 , t 3 , t 4 is the rational expression k is not symmetric in t 1 , t 2 , t 3 , t 4 ; by permuting the indeterminates, k can take 6 different values The maps k → 1/k and k → 1 − k generate the anharmonic group of order 6, which acts regularly on the 6 values of the cross-ratio.It is straightforward to check that the rational expression Hence, u is a symmetric rational function of t 1 , t 2 , t 3 , t 4 .Assume that t 1 , t 2 , t 3 , t 4 are roots of the quartic polynomial Then u can be expressed by the coefficients α 0 , . . ., α 4 as It may be observed that if Γ is given in Legendre form then its j-invariant is Assume that j(Γ) = 0, that is, c 2 − c + 1 = 0.Then, the Hessian H of Γ is This shows that H is the union of three nonconcurrent lines whose intersection points are the corners of Γ.
To our further investigation we need the following result.
Proposition 5.1.Let Γ be an irreducible cubic curve in PG(2, K) defined over an algebraically closed field K of characteristic different from 2 and 3.
For each i = 1, . . ., 7, take pairwise distinct nonsingular points P i , Q i , R i ∈ Γ such that the triple {P i , Q i , R i } is collinear.Assume that there exists a point T off Γ such that quadruples {T, P i , Q i , R i } are collinear and that their crossratio (T, P i , Q i , R i ) is a constant κ.Then Γ has j-invariant 0 and T is one of the three corners of Γ.
Proof.We explicitly present the proof for nonsingular cubics and for cubics with a node, as the cuspidal case can be handled with similar, even much simpler, computation.Therefore, Γ has 9 or 3 inflection points.Pick an inflection point off the tangents to Γ through T .Fix a reference frame such that this inflection point is Y ∞ = (0, 1, 0) and that Γ is in Legendre form Then T = (a, b) is an affine point and so are P i , Q i , R i .Let ℓ be the generic line through T with parametric equation x = a + t, y = b + mt.The parameters of the points of ℓ which also lie in Γ, say P, Q, R, are the roots τ, τ ′ , τ ′′ of the cubic polynomial The cross-ratio k of T, P, Q, R is equal to the cross-ratio of 0, τ, τ ′ , τ ′′ , and its u(k) value can be computed from (1) by substituting From this, u(k) = f (m) 3 /g(m) 2 , where f, g have m-degree 2 and 3, respectively.Let m i be the slope of the line containing the points T, P i , Q i , R i (i = 1, . . ., 7).By our assumption, which implies f (m) 3 g(m) 2 = u(κ) for all m, and this holds true even if one of the m i 's is infinite.If the rational function f (m) 3 g(m) 2 is constant then either its derivative is constant zero, or f (m) ≡ 0, or g(m) ≡ 0. In any of these cases, By setting If c(c − 1) = 0 then β 0 = 2b = 0 and β 1 = 2(a − c) = 0, which implies T ∈ Γ, a contradiction.This proves c 2 − c + 1 = 0 and j(Γ) = 0 by (2).Moreover, therefore by (3), the lines X = a and Y = b through T are components of the Hessian curve of Γ.This shows that T is a corner point of Γ.
Theorem 5.2.Under the hypotheses of Proposition 5.1, (i) the affine reference frame in PG(2, K) can be chosen such that Γ has affine equation X3 + Y 3 = 1 and that T = (0, 0); (ii) there is a perspectivity u of order 3 with center T leaving Γ invariant; (iii) the constant cross-ratio κ is a root of the equation X 2 − X + 1 = 0.
Therefore the following result holds.
Theorem 5.3.Let K be an algebraically closed field of characteristic different from 2 and 3. Let Λ = (Λ 1 , Λ 2 , Λ 3 ) be a dual 3-net of order n ≥ 7 embedded in PG(2, K) which lies on an irreducible cubic curve Γ.If Γ is either singular or is nonsingular with j(Γ) = 0 then Λ is not in perspective position.If j(Γ) = 0 then there are at most three points T 1 , T 2 , T 3 such that Λ is in perspective position with center T i .
Proof.Let T be a point such that Λ is in perspective position with center T .By Proposition 2.3, Λ has a constant cross-ratio κ, hence Theorem 5.2 applies over the algebraic closure of K.
Theorem 5.3 has the following corollary.
Corollary 5.4.Let K be an algebraically closed field of characteristic different from 2 and 3. Then no dual 4-net of order n ≥ 7 embedded in PG(2, K) has a derived dual 3-net lying on a plane cubic.
Finally, we show the existence of proper algebraic dual 3-nets in perspective position.
Example 5.5.Let K be a field of characteristic different from 2 and 3, Γ : X 3 + Y 3 = Z 3 , T = (0, 0, 1) and u : (x, y, z) → (εx, εy, z) with third root of unity ε.The infinite point O(1, −1, 0) is an inflection point of Γ, left invariant by u.Since u leaves Γ invariant as well, u induces an automorphism of the abelian group of (Γ, +, O); we denote the automorphism by u, too.As the line T P contains the points P u , P u 2 for any P ∈ Γ, one has P + P u + P u 2 = O.Let H be a subgroup of (Γ, +) of finite order n such that H u = H.For any P ∈ Γ with P − P u ∈ H, the cosets form a dual 3-net which is in perspective position with center T .Indeed, for any A 1 + P ∈ Λ 1 and A 2 + P u ∈ Λ 2 , the line joining them passes through −A 1 − A 2 + P u 2 ∈ Λ 3 .

Lemma 4 . 1 .
κ = −1 and T is the pole of ℓ in the polarity arising from C.

Proposition 4 .
3 has the following corollary.Corollary 4.4.Let K be an algebraically closed field of characteristic zero or greater than n.Then no dual 4-net of order n embedded in PG(2, K) has a derived dual 3-net of conic-line type.