projective line configurations

1 Introduction

1.1 Definition

A projective line configuration consists of a collectionsof $p$ points and $\mathrm{\ell }$ lines in a projective space such thatthrough each point of the configuration there pass a fixednumber $\lambda$ of lines of the configuration and on eachline of the configuration there are found a fixed number $\pi$of points of the configuration. It is not required that theintersection of any two lines belonging to the configurationbe a line of the configuration nor that through every two pointsof the configuration there pass a line of the configuration.(Indeed, this will usually not be the case except for the verysimplest examples with small values of $\pi$ and $\lambda$.)Besides being interesting geometric objects in their own right,projective configurations arise naturally in geometricdefinitions, constructions, and theorems, and also in suchcontexts as collections of special points and lines associatedto algebraic varieties.

1.2 Examples

A simple example of a projective configuration is a triangle:

$$,(0,0);(30,30)**\mathrm{@}-,(14,17)*b,(22,30);(52,0)**\mathrm{@}-,(38,17)*a,(0,5);(52,5)**\mathrm{@}-,(26,2)*c,(5,5)*\bullet ,(5,2)*A,(47,5)*\bullet ,(46,2)*B,(26,26)*\bullet ,,(30,26)*C$$

Here we have 3 points, $A$,$B$,$C$ and 3 lines, $a$,$b$,$c$with the property that each of the points lines on exactly2 of the lines ($A$ lies on $b$ and $c$, $B$ lies on $a$and $c$, and $C$ lies on $a$ and $b$) and that each of thelines passes through exactly two of the points ($a$ passesthrough $B$ and $C$, $b$ passes through $A$ and $C$, and$c$ passes through $A$ and $B$.)

Another example is a complete quadrangle. This configurationconsists of 4 points $A$, $B$, $C$, $D$ and the 6 lines $a$,$b$, $c$, $d$, $e$, $f$ as illustrated below:

$$,(5,35)*\bullet ,(8,37)*A,(35,35)*\bullet ,(32,37)*B,(5,5)*\bullet ,(8,2)*C,(35,5)*\bullet ,(32,2)*D,(0,5);(40,5)**\mathrm{@}-,(20,2)*c,(0,35);(40,35)**\mathrm{@}-,(20,37)*a,(5,0);(5,40)**\mathrm{@}-,(2,20)*d,(35,0);(35,40)**\mathrm{@}-,(37,20)*b,(0,0);(40,40)**\mathrm{@}-,(27,30)*f,(0,40);(40,0)**\mathrm{@}-,(12,30)*e$$

Since each of the points lieson exactly 3 of the lines and each the lines contains exactlytwo of the points, this is indeed a bona fide projectiveconfiguration. (Note that the intersection of the lines $e$and $f$ is not highlighted as this is not one of the points ofthe configurations — as mentioned at top, not every intersectionof two lines of the configuration need be a point of theconfiguration.)

1.3 Notation

When discussing line configurations, use is made of thenotation $(p_{\lambda }{\mathrm{\ell }}_{\pi })$ orto indicate that the configuration contains $p$ points and$\mathrm{\ell }$ lines with $\lambda$ lines passing through eachpoint and $\pi$ points on each line. Thus, we would saythat the triangle is a configuration of type $(3_2{3}_2)$(or ) andthe complete quadrangle is a configuration of type$(4_3,6_2)$.

It is worth pointing out that the four numbers $p$, $\mathrm{\ell }$,$\pi$, $\lambda$ are linked by the relation $p\cdot \lambda =\pi \cdot \mathrm{\ell }$. The reason for this is a counting argument —we could count pairs consisting of a point and a line passingthrough that point two ways. We could start with the $p$ pointsand count $\lambda$ lines passing through each point to arriveat $p\lambda$ pairs. Alternatively, we could start with the$\mathrm{\ell }$ lines and count $\pi$ points on each line to arriveat $\pi \mathrm{\ell }$ pairs. Since we are counting the same objects(pairs consisting of incident points and lines), we mustarrive at the same number either way.

Finally, when dealing with cases where $p=\mathrm{\ell }$ (and hence,by what was said above, $\pi =\lambda$), one may use theabbreviated notation $(p_{\lambda })$. Thus, one could sayinstead that the triangle is a configuration of type $(3_2)$.Thus notation is most commonly encountered in the contextof self-dual configurations (which will be defined in thenext subsection).

1.4 Choice of Projective Space

So far, we have only spoken of projective configurations in ageneral manner. To discuss the matter in more detail, we needto take into account the projective space within which ourconfiguration is situated.

The need for doing so is rather well illustrated by the factthat certain configurations may not exist in all projectivespaces. As an example, we may consider the Hesse configuration,which is a configuration of type $(9_4{12}_3)$. In thisconfiguration, if we label the points by the letters $A$ through$I$ suitably, the lines pass through the following triplets ofpoints: $ABC$, $DEF$, $GHI$, $ADG$, $BEH$, $CFI$, $AEI$, $BFG$,$CDH$, $AFH$, $BDI$, $CEG$. It is not possible to find 9 pointsand 12 lines in the real projective plane $ℝ{\mathbb{P}}^2$(or, for that matter, any real projective space $ℝ{\mathbb{P}}^n$) which form such a configuration. However, sucha configuration can be found in the complex projective plane$ℂ{\mathbb{P}}^2$ — for instance we could take the9 points with homogeneous coordinates

and the 12 lines with equations

where $r_{\pm }=(1\pm i\sqrt{3})/2$. That the appropriate pointslie on the appropriate lines may be readily verified by a computationwhich is especially effortless if one makes use of the facts that$r_{+}^3=r_{-}^3=1$ and $r_{+}{r}_{-}=1$.

This notion of certain configurations being only found in certainspaces may be clarified by an intrinsic/extrinsic approach. Definean abstract line configuration of type $(p_{\lambda }{\mathrm{\ell }}_{\pi })$to be a triplet $⟨P,L,I⟩$, where $P$ and $Q$ are setsand $I$ is a relation on $P\times L$ such that the followingconditions hold:

• The cardinality of $P$ is $p$.

• The cardinality of $L$ is $\mathrm{\ell }$.

• For every $x\in P$, the cardinality of$\{y\in L\mid I(x,y)\}$ is $\lambda$.

• For every $x\in L$, the cardinality of$\{y\in P\mid I(y,x)\}$ is $\pi$.

• Given two distinct elements $x,y$ of $P$, there exists atmost one element $z$ of $L$ such that $I(x,z)$ and $I(y,z)$.

• Given two distinct elements $x,y$ of $L$, there exists atmost one element $z$ of $P$ such that $I(z,x)$ and $I(z,y)$.

Given a projective space $S$, we may then define an embeding of anabstract line complex $⟨P,L,I⟩$ to be an assignmentof a point of $S$ to every element of $P$ and a line to every elementof $L$ in such a way that the point assigned to an element $x$ of$P$ will lie on the line assigned to an element $y$ of $L$ if andonly if $I(x,y)$. For instance, returning to our first example above,the abstract configuration of the triangle is

$$⟨\{A,B,C\},\{a,b,c\},\{(A,b),(A,c),(B,a),(B,c),(C,a),(C,b)⟩.$$

Not only is this way of thinking useful conceptually,but, as we shall see in the next section, it is useful in practisebecause it lets us divide the work of finding configurations into acombinatorial task of determining abstract configurations and ageometric task of determining which abstract configurations may beembedded in which space.

When the space within which our configuration is embedded is two-dimensional, i.e. happens to be a projective plane, then we canapply the duality operation to obtain another line configurationin which the role of lies and points has been interchanged. Wecall this new configuration the dual of the original configuration.If it happens that the dual configuration is projectively equivalentto the original configuration, we call it a self-dual configuration.For instance, the triangle is a self-dual configuration in$ℝ{\mathbb{P}}^2$, given two triangles, there will be acollineation which maps one into the other. In the example of thecomplete quadrangle, its dual is a configuration known as thecomplete quadrangle, which consists of four lines such that eachpair of lines intersects in on of the six points of the configuration.

$$,(6,6)*\bullet ,(8,4)*C,(26,66)*\bullet ,(29,66)*E,(66,26)*\bullet ,(66,29)*F,(42,18)*\bullet ,(45,17)*D,(18,42)*\bullet ,(16,45)*B,(36,36)*\bullet ,(39,39)*A,(4,0);(28,72)**\mathrm{@}-,(26,11)*c,(0,4);(72,28)**\mathrm{@}-,(10,26)*d,(72,24);(0,48)**\mathrm{@}-,(52,34)*b,(24,72);(48,0)**\mathrm{@}-,(34,52)*a$$

This notion of duality can be extended to abstract configurations.Given an abstract line configuration $⟨P,L,I⟩$, itsdual is $⟨L,P,I^{\prime }⟩$ where $I^{\prime }(x,y)$ if and only if$I(y,x)$. For instance, the abstract configuration of the completequadrangle is

$⟨$$\mathrm{\{}A,B,C,D\},\{a,b,c,d,e,f\},$

$\mathrm{\{}(A,a),(A,d),(A,e),(B,a),(B,b),(B,f),(C,c),(C,d),(C,f),(D,b),(D,c),(D,e)\}⟩$

so its dual is

$⟨$$\mathrm{\{}a,b,c,d,e,f\},\{A,B,C,D\},$

$\mathrm{\{}(a,A),(d,A),(e,A),(a,B),(b,B),(f,B),(c,C),(d,C),(f,C),(b,D),(c,D),(e,D)\}⟩$

which corresponds to the complete quadrilateral.Likewise, we can define a notion of self-duality at the abstractlevel. We will say that an abstract line configuration $⟨P,L,I⟩$ is self-dual if there exists a one-to-onecorrespondence $g:P\to L$ such that, for all $x,y\in P$,we have $I(x,g(y))$ if and only if $I(y,g(x))$.

1.5 Symmetries

Next, we consider the effect of collineations on configurations.Given a configuration in a projective space, a collineation ofthat space will map that configuration into some configuration.If it happens to be mapped into the same configuration, thenwe say that the collineation is a symmetry of the configurationin question.

To illustrate these ideas, let us consider a triangle in $ℝ{\mathbb{P}}^2$ consisting of the points $(1,0,0)$, $(0,1,0)$, $(0,0,1)$and the lines $x=0$, $y=0$, $z=0$. Firstly, consider thecollineation

$x$$\mapsto x+y$

$y$$\mapsto y$

$z$$\mapsto z$

Under this mapping, the lines and points of our configurationtransform as follows:

Since the point $(-1,1,0)$ does not belong to the original configuration,this transform is not a symmetry of our triangle.

Secondly, consider the transformation

$x$$\mapsto y$

$y$$\mapsto z$

$z$$\mapsto x$

Under this mapping, the lines and points of our configurationtransform as follows:

Since the images of the points are points of the original configuration andthe images of the lines also belong to the original configuration, thiscollineation is a symmetry of the configuration.

Thirdly, consider the collineation

$x$$\mapsto 2x$

$y$$\mapsto y$

$z$$\mapsto z$

Under this mapping, the lines and points of our configurationtransform as follows:

Since the images of the points are points of the original configuration andthe images of the lines also belong to the original configuration, thiscollineation is a symmetry of the configuration. (Remember that, since weare dealing with homogeneous coordinates on projective space, overall scalingsdo not matter, so $(1,0,0)$ and $(2,0,0)$ label the same point, likewise$x=0$ and $2x=0$ describe the same line.) Note that this symmetrydiffers from the one in the previous example because each point and line isindividually left invariant as opposed to only having the set of all pointsand the set of all lines be left invariant.

We may also consider permuting the points and lines in abstract lineconfigurations. Given an abstract line configuration $⟨P,L,I⟩$, we will define a symmetry of this configuration to be a pairof permutations $f:P\to P$ and $g:L\to L$ such that, forall $x\in P$ and all $y\in L$, we have $I(f(x),g(y))$ if and only if$I(x,y)$.

We may relate these abstract and concrete symmetry groups as follows.Suppose that we have an abstract configuration $C$ which is embeddedin a projective space $P$ as a configuration $C^{\prime }$. Let $G_a$ be thegroup of symmetries of $C$. Let $G_e$ be the group of collineationsof $P$ which preserves $C^{\prime }$ and let $G_f$ be the group of collineationswhich leaves the points and lines of $C^{\prime }$ fixed individually. Then$G_f$ is a normal subgroup of $G_e$ and the quotient group $G_e/G_f$is a subgroup of $G_a$.

To illustrate how this works, we will consider the symmetry groupsassociated to the example of the triangle studied above. We beginwith $G_e$. Writing down the effect of a linear transform andasking that it preserve the configuration, we find that, in orderfor a transformation to preserve the configuration, it should haveone of the following forms:

Here, $\alpha ,\beta ,\gamma$ are arbitrary non-zero real numbers.If we instead ask that the points and lines of the triangle be fixedindividually, only transforms of the form

$$\{\begin{array}{c}\hfill x\mapsto \alpha x\hfill \\ \hfill y\mapsto \beta y\hfill \\ \hfill z\mapsto \gamma z\hfill \end{array}$$

remain. These form the group $G_f$. Examining the abstractconfiguration of the triangle, we may verify that the followingpermutations are the ones which preserve incidence:

The group presented above is isomorphic to $S_3$. From what was describedearlier, one may also check that $G_e/G_f$ is also isomorphic to $S_3$ so,in this case, $G_e/G_f$ is isomorphic to the whole of $G_a$.

As another illustrative example of symmetry groups of configurations, weshall consider the configuration of type $(4_1{1}_4)$ in $ℝ{\mathbb{P}}^2$consisting of a line $a$ with equation $z=0$ and four points $A$, $B$, $C$,$D$with coordinates $(1,0,0)$, $(1,1,0)$, $(1,2,0)$, $(1,3,0)$ respectively.

$$,(0,5);(50,5)**\mathrm{@}-;(25,7)*a,(10,5)*\bullet ,(10,2)*A,(20,5)*\bullet ,(20,2)*B,(30,5)*\bullet ,(30,2)*C,(40,5)*\bullet ,(40,2)*D$$

In order to preserve this configuration, a collineation must have oneof the following forms:

Here $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers with neither$\alpha$ nor $\beta$ equal to zero. These transforms form the group $G_e$.Of these, the transforms

$$\{\begin{array}{c}\hfill x\mapsto \alpha x+\beta z\hfill \\ \hfill y\mapsto \alpha y+\gamma z\hfill \\ \hfill z\mapsto \delta z\hfill \end{array}$$

fix the points individually so form the normal subgroup $G_f$. As forsymmetries of the abstract configuration, since there is only one line, $g$is trivial whilst $f$ can be any permutation of $4$ objects because theonly relation to be preserved is that all $4$ points line on the same line.Hence, $G_a$ is isomorphic to $S_4$. However, $G_e/G_f$ is isomorphicto the Klein viergruppe, so here we have a case in which $G_e/G_f$ is aproper subgroup of $G_a$.

1.6 Generalizations

In projective spaces of dimension higher than two, we can considerconfigurations consisting not only of points and lines but also ofhigher-dimensional subspaces. For instance, in three or moredimensions, we can consider configurations consisting of points,lines, and planes. Specifically, such a configuration consists ofa set of $n_{00}$ points, $n_{11}$ lines, and $n_{22}$ planes suchthat each point lies on exactly $n_{01}$ lines and $n_{02}$ planes,each line contains $n_{10}$ points and lies on $n_{12}$ planes, andeach plane contains exactly $n_{20}$ points and $n_{21}$ planes,where $n_{00}$, $n_{01}$, $n_{10}$, $n_{02}$, $n_{11}$, $n_{20}$,$n_{12}$, $n_{21}$, and $n_{22}$ are positive integers. Anexample of such an object consists of the four points, six lines,and four planes which comprise the vertices, edges, and faces ofa tetrahedron. Other than mentioning that there exists such ageneralization, we shall not pursue this topic further here, butshall confine our attention to configurations consisting only ofpoints and lines in this article.

2 Determination

2.1 Introduction

Having described the general theory of line configurations, we nowturn our attention to the determination of configurations. Followingthe methodology described above, we will proceed in two steps, firstdetermining abstract configurations, then studying their embeddingsin projective spaces.

2.2 Restrictions on $p$, $\mathrm{\ell }$, $\pi$, $\lambda$

We will begin by deriving some conditions which limit the possiblevalues of $p$, $\mathrm{\ell }$, $\pi$, $\lambda$ which can occur for aline configuration. Already, we have noted one such restrictionabove, namely $p\cdot \lambda =\pi \cdot \mathrm{\ell }$.

Because there must be at least as many points as there are pointson any line, we must have $p\ge \pi$. Likewise, because theremust be at least as many lines as pass through any point, we mustalso have $\mathrm{\ell }\ge \lambda$.

Let $P$ be any point of the configuration. Then there will be$\lambda$ lines passing through $P$, each of which will passthrough $\lambda -1$ points in addition to $P$. Since a lineis determined by two points and all $\lambda$ lines have $P$ incommon, no two of them will have any other point in common, hencethere will be $\lambda (\pi -1)$ distinct points located on theselines. Thus, we conclude that $p\ge \lambda (\pi -1)$. Byinterchanging “point” and “line” in the argument just given,we conclude that $\mathrm{\ell }\ge \pi (\lambda -1)$.

Suppose that $\pi \ge 2$. Since at most one line of the configurationcan pass through two points but every line of the configuration mustpass through at least two points of the configuration, there can be nomore lines in the configuration than there are pairs of points, so$\mathrm{\ell }\le \left(\genfrac{}{}{0pt}{}{p}{2}\right)$. Dually, we must have $p\le \left(\genfrac{}{}{0pt}{}{\mathrm{\ell }}{2}\right)$.

As an illustration of these conditions, we will now ask what limitationsthey impose on the types of configurations which have 12 points. Fromthe inequality $p\ge \pi$, we see that $\pi$ is limited to the values1,2,3,4,5,6,7,8,9,10,11,12.

If $\pi =1$, then we have $\mathrm{\ell }=12\lambda$, hence the possible typesare $({12}_{\lambda }12{\lambda }_1)$. All the other inequalities aresatisfied or irrelevant and, as we shall see, for every choice of $\lambda$,there is a configuration of this type.

Likewise, if $\lambda =1$, then we have $\mathrm{\ell }\pi =12$. Thus, we havethe possibilities $({12}_1{12}_1)$, $({12}_1{6}_2)$, $({12}_1{4}_3)$, $({12}_1{3}_4)$,$({12}_1{2}_6)$, and $({12}_1{1}_{1}2)$, all of which satisfy the remaininginequalities and which happen to occur as types of configurations.

We now focus our attention to the cases where $\pi \ge 2$ and $\lambda \ge 2$.Then we have the inequalities $\mathrm{\ell }\le \left(\genfrac{}{}{0pt}{}{p}{2}\right)=66$ and$\left(\genfrac{}{}{0pt}{}{\mathrm{\ell }}{2}\right)\ge p=12$ to reckon with. These limit $\mathrm{\ell }$ to thevalues $6\le \mathrm{\ell }\le 66$. When $\lambda \ge 2$, the inequality$p\ge \lambda (\pi -1)$ implies that $7\ge \pi$, so it turns out thatthe possible values 8,9,10,11,12 mentioned above are ruled out.Summarrizing, in this case we have the following restrictions on the ranges ofour constants:

$6\le$$\mathrm{\ell }\le 66$

$2\le$$\lambda \le \mathrm{\ell }$

$2\le$$\pi \le 7$

To finish, we will consider the remaining possible values of $\pi$one at a time. When $\pi =2$, we have $\mathrm{\ell }=6\lambda$.Thus, the inequality $\mathrm{\ell }\ge \pi (\lambda -1)$ is automaticallysatisfied and the inequality $p\ge \lambda (\pi -1)$ reduces to$\lambda \le 12$. We have the following possibilities:

When $\pi =3$, we have $\mathrm{\ell }=4\lambda$. Again, the inequality$\mathrm{\ell }\ge \pi (\lambda -1)$ is automatically satisfied. Theinequality $p\ge \lambda (\pi -1)$ reduces to $6\ge \lambda$,hence we have the following possibilities:

When $\pi =4$, we have $\mathrm{\ell }=3\lambda$. Then, the inequality$p\ge \lambda (\pi -1)$ becomes $4\ge \lambda$ and the inequality$\mathrm{\ell }\ge \pi (\lambda -1)$ also becomes $4\ge \lambda$, hence wehave the following possibilities:

When $\pi =5$, we have $12\lambda =5\mathrm{\ell }$. Since 5 and 12 arecoprime, this implies that $\lambda =5n$ and $\mathrm{\ell }=12n$ forsome positive integer $n$. But then, the inequality $p\ge \lambda (\pi -1)$ would become $12\ge 20n$, which is impossible because$n>0$, so we have no configurations with $p=12$ and $\pi =5$.

When $\pi =6$, we have $\mathrm{\ell }=2\lambda$. Then, the inequality$p\ge \lambda (\pi -1)$ becomes $12\ge 5\lambda$, so only$\lambda =2$ is possible. Thus, we must also have $\mathrm{\ell }=4$.However, the inequality $\mathrm{\ell }\ge \pi (\lambda -1)$ is not satisfiedwhen $\mathrm{\ell }=4$, $\pi =6$, and $\lambda =2$, so we have noconfigurations with $p=12$ and $\pi =6$.

When $\pi =7$, we have $12\lambda =7\mathrm{\ell }$. Since 7 and 12 arecoprime, this implies that $\lambda =7n$ and $\mathrm{\ell }=12n$ forsome positive integer $n$. But then, the inequality $p\ge \lambda (\pi -1)$ would become $12\ge 42n$, which is impossible because$n>0$, so we have no configurations with $p=12$ and $\pi =7$.

2.3 The Cases $\lambda =1$ and $\pi =1$

Having deduced and illustrated limitations on the four constants$p$, $\mathrm{\ell }$, $\pi$, $\lambda$, we now turn our attention to thedetermining which sets of numbers describe actual configurations.We begin with the easy case where one or both of $\lambda$ and$\pi$ equals $1$.

Before proceeding further, it is worth pointing out that we areonly interested in classifying abstract configurations up toequivalence by permutation. That is to say, we will considertwo abstract configurations $⟨P,L,I⟩$ and$⟨P^{\prime },L^{\prime },I^{\prime }⟩$ equivalent if there exist one-to-onecorrespondences $f:P\to P^{\prime }$ and $g:L\to L^{\prime }$ suchthat $I(x,y)$ if and only if $I^{\prime }(f(x),g(y))$. The reason fordoing this is that, since it is easy enough to permute elementsin a given configuration, listing only one configuration out ofan equivalence class cuts down on the number.

Suppose that we have an abstract configuration with $\pi =1$.Then, to every element of $L$ we may associate exactly oneelement of $P$. Furthermore, we may define an equivalencerelation on $L$ by equating the lines which pass through thesame point. Thus, our lines are partitioned into $p$ partitionsof $\lambda$ points each. Conversely, given two numbers $\lambda$and $p$ and setting $\mathrm{\ell }=p\lambda$, we can make aconfiguration by taking a set $L$ with $\mathrm{\ell }$ elements andpartitioning it into $p$ subsets of $\lambda$ elements each,then associating to each equivalence class an element of theset $P$.

3 Catalogue

[Under Construction]

Title	projective line configurations
Canonical name	ProjectiveLineConfigurations
Date of creation	2013-03-22 18:23:45
Last modified on	2013-03-22 18:23:45
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	55
Author	rspuzio (6075)
Entry type	Topic
Classification	msc 14N20
Classification	msc 14N10
Classification	msc 05B99
Classification	msc 52C30
Classification	msc 51N15
Classification	msc 51E20
Classification	msc 51A45
Classification	msc 51A05
Classification	msc 51A20
Synonym	projective configuration
Synonym	line configuration