projective line configurations

projective line configurations

1

Introduction

1.1

Definition

A

projective line configuration

consists of a

collections

of

p

points and

ℓ

lines in a

projective space

such that
through each point of the

configuration

there pass a fixed
number

λ

of lines of the configuration and on each
line of the configuration there are found a fixed number

π

of points of the configuration. It is not required that the

intersection

of any two lines belonging to the configuration
be a line of the configuration nor that through every two points
of the configuration there pass a line of the configuration.
(Indeed, this will usually not be the case except for the very
simplest examples with small values of

π

and

λ

.)
Besides being interesting geometric objects in their own right,
projective configurations arise naturally in geometric
definitions, constructions, and

theorems

, and also in such

contexts

as collections of special points and lines associated
to

algebraic varieties

.

1.2

Examples

A simple example of a projective configuration is a

triangle

:

{xy}

,

(

0

,

0

)

;

(

30

,

30

)

*

*

@

-

,

(

14

,

17

)

*

b

,

(

22

,

30

)

;

(

52

,

0

)

*

*

@

-

,

(

38

,

17

)

*

a

,

(

0

,

5

)

;

(

52

,

5

)

*

*

@

-

,

(

26

,

2

)

*

c

,

(

5

,

5

)

*

∙

,

(

5

,

2

)

*

A

,

(

47

,

5

)

*

∙

,

(

46

,

2

)

*

B

,

(

26

,

26

)

*

∙

,

,

(

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 exactly
2 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 the
lines

passes through

exactly two of the points (

a

passes
through

B

and

C

,

b

passes through

A

and

C

, and

c

passes through

A

and

B

.)

Another example is a

complete quadrangle

. This configuration
consists of 4 points

A

,

B

,

C

,

D

and the 6 lines

a

,

b

,

c

,

d

,

e

,

f

as illustrated below:

{xy}

,

(

5

,

35

)

*

∙

,

(

8

,

37

)

*

A

,

(

35

,

35

)

*

∙

,

(

32

,

37

)

*

B

,

(

5

,

5

)

*

∙

,

(

8

,

2

)

*

C

,

(

35

,

5

)

*

∙

,

(

32

,

2

)

*

D

,

(

0

,

5

)

;

(

40

,

5

)

*

*

@

-

,

(

20

,

2

)

*

c

,

(

0

,

35

)

;

(

40

,

35

)

*

*

@

-

,

(

20

,

37

)

*

a

,

(

5

,

0

)

;

(

5

,

40

)

*

*

@

-

,

(

2

,

20

)

*

d

,

(

35

,

0

)

;

(

35

,

40

)

*

*

@

-

,

(

37

,

20

)

*

b

,

(

0

,

0

)

;

(

40

,

40

)

*

*

@

-

,

(

27

,

30

)

*

f

,

(

0

,

40

)

;

(

40

,

0

)

*

*

@

-

,

(

12

,

30

)

*

e

Since each of the points lies
on exactly 3 of the lines and each the lines contains exactly
two of the points, this is indeed a bona fide projective
configuration. (Note that the intersection of the lines

e

and

f

is not highlighted as this is not one of the points of
the configurations — as mentioned at top, not every intersection
of two lines of the configuration need be a point of the
configuration.)

1.3

Notation

When discussing line configurations, use is made of the
notation

(

p

λ

⁢

ℓ

π

)

or

(

p

λ

π

ℓ

)

to indicate that the configuration contains

p

points and

ℓ

lines with

λ

lines passing through each
point and

π

points on each line. Thus, we would say
that the triangle is a configuration of type

(

3

2

⁢

3

2

)

(or

(

3

2

2

3

)

) and
the complete quadrangle is a configuration of type

(

4

3

,

6

2

)

.

It is worth pointing out that the four numbers

p

,

ℓ

,

π

,

λ

are linked by the

relation

p

⋅

λ

=

π

⋅

ℓ

. The reason for this is a counting

argument

—
we could count pairs consisting of a point and a line passing
through that point two ways. We could start with the

p

points
and count

λ

lines passing through each point to arrive
at

p

⁢

λ

pairs. Alternatively, we could start with the

ℓ

lines and count

π

points on each line to arrive
at

π

⁢

ℓ

pairs. Since we are counting the same objects
(pairs consisting of

incident

points and lines), we must
arrive at the same number either way.

Finally, when dealing with cases where

p

=

ℓ

(and hence,
by what was said above,

π

=

λ

), one may use the
abbreviated notation

(

p

λ

)

. Thus, one could say
instead that the triangle is a configuration of type

(

3

2

)

.
Thus notation is most commonly encountered in the context
of self-dual configurations (which will be defined in the
next subsection).

1.4

Choice of Projective Space

So far, we have only spoken of projective configurations in a
general manner. To discuss the matter in more detail, we need
to take into account the projective space within which our
configuration is situated.

The need for doing so is rather well illustrated by the fact
that certain configurations may not exist in all projective
spaces. As an example, we may consider the

Hesse configuration

,
which is a configuration of type

(

9

4

⁢

12

3

)

. In this
configuration, if we label the points by the letters

A

through

I

suitably, the lines pass through the following

triplets

of
points:

A

⁢

B

⁢

C

,

D

⁢

E

⁢

F

,

G

⁢

H

⁢

I

,

A

⁢

D

⁢

G

,

B

⁢

E

⁢

H

,

C

⁢

F

⁢

I

,

A

⁢

E

⁢

I

,

B

⁢

F

⁢

G

,

C

⁢

D

⁢

H

,

A

⁢

F

⁢

H

,

B

⁢

D

⁢

I

,

C

⁢

E

⁢

G

. It is not possible to find 9 points
and 12 lines in the real projective plane

ℝ

⁢

ℙ

2

(or, for that matter, any real projective space

ℝ

⁢

ℙ

n

) which form such a configuration. However, such
a configuration can be found in the

complex projective plane

ℂ

⁢

ℙ

2

— for

instance

we could take the
9 points with

homogeneous coordinates

A

:

(

1

,

-

r

+

,

0

)

B

:

(

1

,

-

1

,

0

)

C

:

(

1

,

-

r

-

,

0

)

D

:

(

-

r

+

,

0

,

1

)

E

:

(

-

1

,

0

,

1

)

F

:

(

-

r

-

,

0

,

1

)

G

:

(

0

,

1

,

-

r

+

)

H

:

(

0

,

1

,

-

1

)

I

:

(

0

,

1

,

-

r

-

)

and the 12 lines with

equations

A

⁢

B

⁢

C

:

⁢

z

=

0

D

⁢

E

⁢

F

:

⁢

y

=

0

G

⁢

H

⁢

I

:

⁢

x

=

0

A

⁢

D

⁢

G

:

⁢

x

+

r

-

⁢

y

+

r

+

⁢

z

=

0

B

⁢

E

⁢

H

:

⁢

x

+

y

+

z

=

0

C

⁢

F

⁢

I

:

⁢

x

+

r

+

⁢

y

+

r

-

⁢

z

=

0

A

⁢

E

⁢

I

:

⁢

x

+

r

-

⁢

y

+

z

=

0

B

⁢

F

⁢

G

:

⁢

x

+

y

+

r

-

⁢

z

=

0

C

⁢

D

⁢

H

:

⁢

r

-

⁢

x

+

y

+

z

=

0

A

⁢

F

⁢

H

:

⁢

r

+

⁢

x

+

y

+

z

=

0

B

⁢

D

⁢

I

:

⁢

x

+

y

+

r

+

⁢

z

=

0

C

⁢

E

⁢

G

:

⁢

x

+

r

+

⁢

y

+

z

=

0

,

where

r

±

=

(

1

±

i

⁢

3

)

/

2

. That the appropriate points
lie on the appropriate lines may be readily verified by a computation
which 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 certain
spaces may be clarified by an intrinsic/extrinsic approach. Define
an

abstract line configuration

of type

(

p

λ

⁢

ℓ

π

)

to be a triplet

⟨

P

,

L

,

I

⟩

, where

P

and

Q

are sets
and

I

is a relation on

P

×

L

such that the following
conditions hold:

•

The

cardinality

of

P

is

p

.

•

The cardinality of

L

is

ℓ

.

•

For every

x

∈

P

, the cardinality of

{

y

∈

L

∣

I

⁢

(

x

,

y

)

}

is

λ

.

•

For every

x

∈

L

, the cardinality of

{

y

∈

P

∣

I

⁢

(

y

,

x

)

}

is

π

.

•

Given two distinct

elements

x

,

y

of

P

, there exists at
most one element

z

of

L

such that

I

⁢

(

x

,

z

)

and

I

⁢

(

y

,

z

)

.

•

Given two distinct elements

x

,

y

of

L

, there exists at
most 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 an
abstract line complex

⟨

P

,

L

,

I

⟩

to be an assignment
of a point of

S

to every element of

P

and a line to every element
of

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 and
only 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 practise
because it lets us divide the work of finding configurations into a
combinatorial task of determining abstract configurations and a
geometric task of determining which abstract configurations may be
embedded 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 can
apply the

duality

operation

to obtain another line configuration
in which the role of lies and points has been interchanged. We
call this new configuration the dual of the original configuration.
If it happens that the dual configuration is

projectively equivalent

to the original configuration, we call it a self-dual configuration.
For instance, the triangle is a self-dual configuration in

ℝ

⁢

ℙ

2

, given two triangles, there will be a

collineation

which maps one into the other. In the example of the
complete quadrangle, its dual is a configuration known as the
complete quadrangle, which consists of four lines such that each
pair of lines intersects in on of the six points of the configuration.

{xy}

,

(

6

,

6

)

*

∙

,

(

8

,

4

)

*

C

,

(

26

,

66

)

*

∙

,

(

29

,

66

)

*

E

,

(

66

,

26

)

*

∙

,

(

66

,

29

)

*

F

,

(

42

,

18

)

*

∙

,

(

45

,

17

)

*

D

,

(

18

,

42

)

*

∙

,

(

16

,

45

)

*

B

,

(

36

,

36

)

*

∙

,

(

39

,

39

)

*

A

,

(

4

,

0

)

;

(

28

,

72

)

*

*

@

-

,

(

26

,

11

)

*

c

,

(

0

,

4

)

;

(

72

,

28

)

*

*

@

-

,

(

10

,

26

)

*

d

,

(

72

,

24

)

;

(

0

,

48

)

*

*

@

-

,

(

52

,

34

)

*

b

,

(

24

,

72

)

;

(

48

,

0

)

*

*

@

-

,

(

34

,

52

)

*

a

This notion of duality can be extended to abstract configurations.
Given an abstract line configuration

⟨

P

,

L

,

I

⟩

, its
dual is

⟨

L

,

P

,

I

′

⟩

where

I

′

⁢

(

x

,

y

)

if and only if

I

⁢

(

y

,

x

)

. For instance, the abstract configuration of the complete
quadrangle is

⟨

{

A

,

B

,

C

,

D

}

,

{

a

,

b

,

c

,

d

,

e

,

f

}

,

{

(

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

⟨

{

a

,

b

,

c

,

d

,

e

,

f

}

,

{

A

,

B

,

C

,

D

}

,

{

(

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 abstract
level. We will say that an abstract line configuration

⟨

P

,

L

,

I

⟩

is self-dual if there exists a

one-to-one
correspondence

g

:

P

→

L

such that, for all

x

,

y

∈

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 of
that space will map that configuration into some configuration.
If it happens to be mapped into the same configuration, then
we say that the collineation is a

symmetry

of the configuration
in question.

To illustrate these ideas, let us consider a triangle in

ℝ

⁢

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 the
collineation

x

↦

x

+

y

y

↦

y

z

↦

z

Under this

mapping

, the lines and points of our configuration
transform as follows:

(

1

,

0

,

0

)

↦

(

1

,

0

,

0

)

x

=

0

↦

x

+

y

=

0

(

0

,

1

,

0

)

↦

(

-

1

,

1

,

0

)

y

=

0

↦

y

=

0

(

0

,

0

,

1

)

↦

(

0

,

0

,

1

)

z

=

0

↦

z

=

0

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

↦

y

y

↦

z

z

↦

x

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

(

1

,

0

,

0

)

↦

(

0

,

1

,

0

)

x

=

0

↦

y

=

0

(

0

,

1

,

0

)

↦

(

0

,

0

,

1

)

y

=

0

↦

z

=

0

(

0

,

0

,

1

)

↦

(

1

,

0

,

0

)

z

=

0

↦

z

=

0

Since the images of the points are points of the original configuration and
the images of the lines also belong to the original configuration, this
collineation is a symmetry of the configuration.

Thirdly, consider the collineation

x

↦

2

⁢

x

y

↦

y

z

↦

z

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

(

1

,

0

,

0

)

↦

(

2

,

0

,

0

)

x

=

0

↦

2

⁢

x

=

0

(

0

,

1

,

0

)

↦

(

0

,

1

,

0

)

y

=

0

↦

y

=

0

(

0

,

0

,

1

)

↦

(

0

,

0

,

1

)

z

=

0

↦

z

=

0

Since the images of the points are points of the original configuration and
the images of the lines also belong to the original configuration, this
collineation is a symmetry of the configuration. (Remember that, since we
are dealing with homogeneous coordinates on projective space, overall

scalings

do not matter, so

(

1

,

0

,

0

)

and

(

2

,

0

,

0

)

label the same point, likewise

x

=

0

and

2

⁢

x

=

0

describe the same line.) Note that this symmetry
differs from the one in the previous example because each point and line is
individually left

invariant

as opposed to only having the set of all points
and the set of all lines be left invariant.

We may also consider permuting the points and lines in abstract line
configurations. Given an abstract line configuration

⟨

P

,

L

,

I

⟩

, we will define a symmetry of this configuration to be a pair
of

permutations

f

:

P

→

P

and

g

:

L

→

L

such that, for
all

x

∈

P

and all

y

∈

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 embedded
in a projective space

P

as a configuration

C

′

. Let

G

a

be the
group of symmetries of

C

. Let

G

e

be the group of collineations
of

P

which preserves

C

′

and let

G

f

be the group of collineations
which leaves the points and lines of

C

′

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 groups
associated to the example of the triangle studied above. We begin
with

G

e

. Writing down the effect of a linear transform and
asking that it preserve the configuration, we find that, in order
for a transformation to preserve the configuration, it should have
one of the following forms:

{

x

↦

α

⁢

x

y

↦

β

⁢

y

z

↦

γ

⁢

z

{

x

↦

α

⁢

y

y

↦

β

⁢

z

z

↦

γ

⁢

x

{

x

↦

α

⁢

z

y

↦

β

⁢

x

z

↦

γ

⁢

y

{

x

↦

α

⁢

x

y

↦

β

⁢

z

z

↦

γ

⁢

x

{

x

↦

α

⁢

z

y

↦

β

⁢

y

z

↦

γ

⁢

x

{

x

↦

α

⁢

y

z

↦

β

⁢

y

z

↦

γ

⁢

z

Here,

α

,

β

,

γ

are arbitrary non-zero

real numbers

.
If we instead ask that the points and lines of the triangle be fixed
individually, only transforms of the form

{

x

↦

α

⁢

x

y

↦

β

⁢

y

z

↦

γ

⁢

z

remain. These form the group

G

f

. Examining the abstract
configuration of the triangle, we may verify that the following
permutations are the ones which preserve incidence:

x

A

B

C

f(x)

A

B

C

x

a

b

c

g(x)

a

b

c

x

A

B

C

f(x)

A

C

B

x

a

b

c

g(x)

a

c

b

x

A

B

C

f(x)

B

C

A

x

a

b

c

g(x)

b

c

a

x

A

B

C

f(x)

C

B

A

x

a

b

c

g(x)

c

b

a

x

A

B

C

f(x)

C

A

B

x

a

b

c

g(x)

c

a

b

x

A

B

C

f(x)

B

A

C

x

a

b

c

g(x)

b

a

c

The group presented above is

isomorphic

to

S

3

. From what was described
earlier, 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, we
shall consider the configuration of type

(

4

1

⁢

1

4

)

in

ℝ

⁢

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.

{xy}

,

(

0

,

5

)

;

(

50

,

5

)

*

*

@

-

;

(

25

,

7

)

*

a

,

(

10

,

5

)

*

∙

,

(

10

,

2

)

*

A

,

(

20

,

5

)

*

∙

,

(

20

,

2

)

*

B

,

(

30

,

5

)

*

∙

,

(

30

,

2

)

*

C

,

(

40

,

5

)

*

∙

,

(

40

,

2

)

*

D

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

{

x

↦

α

⁢

x

+

β

⁢

z

y

↦

α

⁢

y

+

γ

⁢

z

z

↦

δ

⁢

z

{

x

↦

α

⁢

x

+

β

⁢

z

y

↦

3

⁢

α

⁢

x

-

α

⁢

y

+

γ

⁢

z

z

↦

δ

⁢

z

{

x

↦

3

⁢

α

⁢

x

-

2

⁢

α

⁢

y

+

β

⁢

z

y

↦

3

⁢

α

⁢

x

-

3

⁢

α

⁢

y

+

γ

⁢

z

z

↦

δ

⁢

z

{

x

↦

3

⁢

α

⁢

x

-

2

⁢

α

⁢

y

+

β

⁢

z

y

↦

6

⁢

α

⁢

x

-

3

⁢

α

⁢

y

+

γ

⁢

z

z

↦

δ

⁢

z

Here

α

,

β

,

γ

,

δ

are real numbers with neither

α

nor

β

equal to zero. These transforms form the group

G

e

.
Of these, the transforms

{

x

↦

α

⁢

x

+

β

⁢

z

y

↦

α

⁢

y

+

γ

⁢

z

z

↦

δ

⁢

z

fix the points individually so form the normal subgroup

G

f

. As for
symmetries of the abstract configuration, since there is only one line,

g

is

trivial

whilst

f

can be any permutation of

4

objects because the
only 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 isomorphic
to the Klein

viergruppe

, so here we have a case in which

G

e

/

G

f

is a

proper subgroup

of

G

a

.

1.6

Generalizations

In projective spaces of

dimension

higher than two, we can consider
configurations consisting not only of points and lines but also of
higher-dimensional

subspaces

. For instance, in three or more
dimensions, we can consider configurations consisting of points,
lines, and planes. Specifically, such a configuration consists of
a set of

n

00

points,

n

11

lines, and

n

22

planes such
that each point lies on exactly

n

01

lines and

n

02

planes,
each line contains

n

10

points and lies on

n

12

planes, and
each 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. An
example of such an object consists of the four points, six lines,
and four planes which comprise the vertices, edges, and faces of
a

tetrahedron

. Other than mentioning that there exists such a

generalization

, we shall not pursue this topic further here, but
shall confine our attention to configurations consisting only of
points and lines in this article.

2

Determination

2.1

Introduction

Having described the general theory of line configurations, we now
turn our attention to the determination of configurations. Following
the methodology described above, we will proceed in two steps, first
determining abstract configurations, then studying their

embeddings

in projective spaces.

2.2

Restrictions on

p

,

ℓ

,

π

,

λ

We will begin by deriving some conditions which limit the possible
values of

p

,

ℓ

,

π

,

λ

which can occur for a
line configuration. Already, we have noted one such

restriction

above, namely

p

⋅

λ

=

π

⋅

ℓ

.

Because there must be at least as many points as there are points
on any line, we must have

p

≥

π

. Likewise, because there
must be at least as many lines as pass through any point, we must
also have

ℓ

≥

λ

.

Let

P

be any point of the configuration. Then there will be

λ

lines passing through

P

, each of which will pass
through

λ

-

1

points in

addition

to

P

. Since a line
is determined by two points and all

λ

lines have

P

in
common, no two of them will have any other point in common, hence
there will be

λ

⁢

(

π

-

1

)

distinct points located on these
lines. Thus, we conclude that

p

≥

λ

⁢

(

π

-

1

)

. By
interchanging “point” and “line” in the argument just given,
we conclude that

ℓ

≥

π

⁢

(

λ

-

1

)

.

Suppose that

π

≥

2

. Since at most one line of the configuration
can pass through two points but every line of the configuration must
pass through at least two points of the configuration, there can be no
more lines in the configuration than there are pairs of points, so

ℓ

≤

(

p

2

)

. Dually, we must have

p

≤

(

ℓ

2

)

.

As an illustration of these conditions, we will now ask what limitations
they impose on the types of configurations which have 12 points. From
the

inequality

p

≥

π

, we see that

π

is limited to the values
1,2,3,4,5,6,7,8,9,10,11,12.

If

π

=

1

, then we have

ℓ

=

12

⁢

λ

, hence the possible types
are

(

12

λ

⁢

12

⁢

λ

1

)

. All the other inequalities are
satisfied or irrelevant and, as we shall see, for every choice of

λ

,
there is a configuration of this type.

Likewise, if

λ

=

1

, then we have

ℓ

⁢

π

=

12

. Thus, we have
the 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 remaining
inequalities and which happen to occur as types of configurations.

We now focus our attention to the cases where

π

≥

2

and

λ

≥

2

.
Then we have the inequalities

ℓ

≤

(

p

2

)

=

66

and

(

ℓ

2

)

≥

p

=

12

to reckon with. These limit

ℓ

to the
values

6

≤

ℓ

≤

66

. When

λ

≥

2

, the inequality

p

≥

λ

⁢

(

π

-

1

)

implies that

7

≥

π

, so it turns out that
the possible values 8,9,10,11,12 mentioned above are ruled out.
Summarrizing, in this case we have the following restrictions on the ranges of
our

constants

:

6

≤

ℓ

≤

66

2

≤

λ

≤

ℓ

2

≤

π

≤

7

To finish, we will consider the remaining possible values of

π

one at a time. When

π

=

2

, we have

ℓ

=

6

⁢

λ

.
Thus, the inequality

ℓ

≥

π

⁢

(

λ

-

1

)

is automatically
satisfied and the inequality

p

≥

λ

⁢

(

π

-

1

)

reduces to

λ

≤

12

. We have the following possibilities:

(

12

2

⁢

12

2

)

(

12

3

⁢

18

2

)

(

12

4

⁢

24

2

)

(

12

5

⁢

30

2

)

(

12

6

⁢

36

2

)

(

12

7

⁢

42

2

)

(

12

8

⁢

48

2

)

(

12

9

⁢

54

2

)

(

12

10

⁢

60

2

)

(

12

11

⁢

66

2

)

When

π

=

3

, we have

ℓ

=

4

⁢

λ

. Again, the inequality

ℓ

≥

π

⁢

(

λ

-

1

)

is automatically satisfied. The
inequality

p

≥

λ

⁢

(

π

-

1

)

reduces to

6

≥

λ

,
hence we have the following possibilities:

(

12

2

⁢

8

3

)

(

12

3

⁢

12

3

)

(

12

4

⁢

16

3

)

(

12

5

⁢

20

3

)

(

12

6

⁢

24

3

)

When

π

=

4

, we have

ℓ

=

3

⁢

λ

. Then, the inequality

p

≥

λ

⁢

(

π

-

1

)

becomes

4

≥

λ

and the inequality

ℓ

≥

π

⁢

(

λ

-

1

)

also becomes

4

≥

λ

, hence we
have the following possibilities:

(

12

2

⁢

6

4

)

(

12

3

⁢

9

4

)

(

12

4

⁢

12

4

)

When

π

=

5

, we have

12

⁢

λ

=

5

⁢

ℓ

. Since 5 and 12 are
coprime, this implies that

λ

=

5

⁢

n

and

ℓ

=

12

⁢

n

for
some positive integer

n

. But then, the inequality

p

≥

λ

⁢

(

π

-

1

)

would become

12

≥

20

⁢

n

, which is impossible because

n

>

0

, so we have no configurations with

p

=

12

and

π

=

5

.

When

π

=

6

, we have

ℓ

=

2

⁢

λ

. Then, the inequality

p

≥

λ

⁢

(

π

-

1

)

becomes

12

≥

5

⁢

λ

, so only

λ

=

2

is possible. Thus, we must also have

ℓ

=

4

.
However, the inequality

ℓ

≥

π

⁢

(

λ

-

1

)

is not satisfied
when

ℓ

=

4

,

π

=

6

, and

λ

=

2

, so we have no
configurations with

p

=

12

and

π

=

6

.

When

π

=

7

, we have

12

⁢

λ

=

7

⁢

ℓ

. Since 7 and 12 are
coprime, this implies that

λ

=

7

⁢

n

and

ℓ

=

12

⁢

n

for
some positive integer

n

. But then, the inequality

p

≥

λ

⁢

(

π

-

1

)

would become

12

≥

42

⁢

n

, which is impossible because

n

>

0

, so we have no configurations with

p

=

12

and

π

=

7

.

2.3

The Cases

λ

=

1

and

π

=

1

Having deduced and illustrated limitations on the four constants

p

,

ℓ

,

π

,

λ

, we now turn our attention to the
determining which sets of numbers describe actual configurations.
We begin with the easy case where one or both of

λ

and

π

equals

1

.

Before proceeding further, it is worth pointing out that we are
only interested in classifying abstract configurations up to
equivalence by permutation. That is to say, we will consider
two abstract configurations

⟨

P

,

L

,

I

⟩

and

⟨

P

′

,

L

′

,

I

′

⟩

equivalent

if there exist one-to-one
correspondences

f

:

P

→

P

′

and

g

:

L

→

L

′

such
that

I

⁢

(

x

,

y

)

if and only if

I

′

⁢

(

f

⁢

(

x

)

,

g

⁢

(

y

)

)

. The reason for
doing this is that, since it is easy enough to permute elements
in a given configuration, listing only one configuration out of
an

equivalence class

cuts down on the number.

Suppose that we have an abstract configuration with

π

=

1

.
Then, to every element of

L

we may associate exactly one
element of

P

. Furthermore, we may define an

equivalence
relation

on

L

by equating the lines which pass through the
same point. Thus, our lines are partitioned into

p

partitions

of

λ

points each.

Conversely

, given two numbers

λ

and

p

and setting

ℓ

=

p

⁢

λ

, we can make a
configuration by taking a set

L

with

ℓ

elements and
partitioning it into

p

subsets of

λ

elements each,
then associating to each equivalence class an element of the
set

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

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