An introduction to Finite Geometry

Geertrui Van de Voorde

Ghent University, Belgium

Pre-ICM International Convention on Mathematical SciencesDelhi

INCIDENCE STRUCTURES

EXAMPLES

I DesignsI GraphsI Linear spacesI Polar spacesI Generalised polygonsI Projective spacesI . . .

Points, vertices, lines, blocks, edges, planes, hyperplanes . . .+ incidence relation

PROJECTIVE SPACES

Many examples are embeddable in a projective space.

V : Vector spacePG(V ): Corresponding projective space

FROM VECTOR SPACE TO PROJECTIVE SPACE

FROM VECTOR SPACE TO PROJECTIVE SPACE

The projective dimension of a projective space is the dimensionof the corresponding vector space minus 1

PROPERTIES OF A PG(V ) OF DIMENSION d

(1) Through every two points, there is exactly one line.

PROPERTIES OF A PG(V ) OF DIMENSION d

(2) Every two lines in one plane intersect, and they intersect inexactly one point.

(3) There are d + 2 points such that no d + 1 of them arecontained in a (d − 1)-dimensional projective spacePG(d − 1, q).

WHICH SPACES SATISFY (1)-(2)-(3)?

THEOREM (VEBLEN-YOUNG 1916)If d ≥ 3, a space satisfying (1)-(2)-(3) is a d-dimensionalPG(V ).

DEFINITIONIf d = 2, a space satisfying (1)-(2)-(3) is called a projectiveplane.

WHICH SPACES SATISFY (1)-(2)-(3)?

THEOREM (VEBLEN-YOUNG 1916)If d ≥ 3, a space satisfying (1)-(2)-(3) is a d-dimensionalPG(V ).

DEFINITIONIf d = 2, a space satisfying (1)-(2)-(3) is called a projectiveplane.

PROJECTIVE PLANES

Points, lines and three axioms

(a) ∀r 6= s ∃!L (b) ∀L 6= M ∃!r (c) ∃r , s, t , u

If Π is a projective plane, then interchanging points and lines,we obtain the dual plane ΠD.

FINITE PROJECTIVE PLANES

DEFINITIONThe order of a projective plane is the number of points on a lineminus 1.

A projective plane of order n has n2 + n + 1 points andn2 + n + 1 lines.

PROJECTIVE SPACES OVER A FINITE FIELD

Fp = Z/Zp if p is primeFq = Fp[X ]/(f (X )), with f (X ) an irreducible polynomial ofdegree h if q = ph, p prime.

NOTATIONV (Fd

q) = V (d , Fq) = V (d , q): vector space in d dimensionsover Fq. The corresponding projective space is denoted byPG(d − 1, q).

PROJECTIVE PLANES OVER A FINITE FIELD

The order of PG(2, q) is q, so a line contains q + 1 points, andthere are q + 1 lines through a point.

PG(2, q) is not the only example of a projective plane, there areother projective planes, e.g. semifield planes.

PROJECTIVE PLANES OVER A FINITE FIELD

The order of PG(2, q) is q, so a line contains q + 1 points, andthere are q + 1 lines through a point.PG(2, q) is not the only example of a projective plane, there areother projective planes, e.g. semifield planes.

WHEN IS A PROJECTIVE PLANE ∼= PG(2, q)?

THEOREMA finite projective plane ∼= PG(2, q) ⇐⇒ Desarguesconfiguration holds for any two triangles that are in perspective.

DESARGUES CONFIGURATION

EXISTENCE AND UNIQUENESS OF A PROJECTIVE PLANE

OF ORDER n

PG(2, q) is an example of a projective plane of order q = ph, pprime.

I Is this the only example of a projective plane of orderq = ph?

I Are there projective planes of order n, where n is not aprime power?

EXISTENCE AND UNIQUENESS OF A PROJECTIVE PLANE

OF ORDER n

PG(2, q) is an example of a projective plane of order q = ph, pprime.

I Is this the only example of a projective plane of orderq = ph?

I Are there projective planes of order n, where n is not aprime power?

THE SMALLEST PROJECTIVE PLANE: PG(2, 2)

The projective plane of order 2, the Fano plane, has:I q + 1 = 2 + 1 = 3 points on a line,I 3 lines through a point.

And it is unique.

THE PROJECTIVE PLANE PG(2, 3)

The projective plane PG(2, 3) has:I q + 1 = 3 + 1 = 4 points on a line,I 4 lines through a point.

And it is unique.

SMALL PROJECTIVE PLANES

The projective planes PG(2, 4), PG(2, 5), PG(2, 7) and PG(2, 8)are unique.

THEOREMThere are 4 non-isomorphic planes of order 9.

THEOREM (BRUCK-CHOWLA-RYSER 1949)Let n be the order of a projective plane, where n ∼= 1 or 2mod 4, then n is the sum of two squares.This theorem rules out projective planes of orders 6 and 14.Is there a projective plane of order 10?

THEOREM (LAM, SWIERCZ, THIEL, BY COMPUTER)There is no projective plane of order 10

SMALL PROJECTIVE PLANES

The projective planes PG(2, 4), PG(2, 5), PG(2, 7) and PG(2, 8)are unique.

THEOREMThere are 4 non-isomorphic planes of order 9.

THEOREM (BRUCK-CHOWLA-RYSER 1949)Let n be the order of a projective plane, where n ∼= 1 or 2mod 4, then n is the sum of two squares.This theorem rules out projective planes of orders 6 and 14.Is there a projective plane of order 10?

THEOREM (LAM, SWIERCZ, THIEL, BY COMPUTER)There is no projective plane of order 10

SMALL PROJECTIVE PLANES

The projective planes PG(2, 4), PG(2, 5), PG(2, 7) and PG(2, 8)are unique.

THEOREMThere are 4 non-isomorphic planes of order 9.

THEOREM (BRUCK-CHOWLA-RYSER 1949)Let n be the order of a projective plane, where n ∼= 1 or 2mod 4, then n is the sum of two squares.This theorem rules out projective planes of orders 6 and 14.Is there a projective plane of order 10?

THEOREM (LAM, SWIERCZ, THIEL, BY COMPUTER)There is no projective plane of order 10

SMALL PROJECTIVE PLANES

The projective planes PG(2, 4), PG(2, 5), PG(2, 7) and PG(2, 8)are unique.

THEOREMThere are 4 non-isomorphic planes of order 9.

THEOREM (BRUCK-CHOWLA-RYSER 1949)Let n be the order of a projective plane, where n ∼= 1 or 2mod 4, then n is the sum of two squares.This theorem rules out projective planes of orders 6 and 14.Is there a projective plane of order 10?

THEOREM (LAM, SWIERCZ, THIEL, BY COMPUTER)There is no projective plane of order 10

OPEN QUESTIONS

I Do there exist projective planes with the order not a primepower?

I How many non-isomorphic projective planes are there of acertain order?

FIRST GEOMETRICAL OBJECTS: SUBSETS

TRIANGLES AND QUADRANGLES IN PROJECTIVE SPACE

I In a projective space, all triangles are "the same".

I In PG(2, q) all quadrangles are "the same".I In PG(3, q), there are two different types of quadrangles:

those contained in a plane, and those not contained in aplane.

TRIANGLES AND QUADRANGLES IN PROJECTIVE SPACE

I In a projective space, all triangles are "the same".

I In PG(2, q) all quadrangles are "the same".

I In PG(3, q), there are two different types of quadrangles:those contained in a plane, and those not contained in aplane.

TRIANGLES AND QUADRANGLES IN PROJECTIVE SPACE

I In a projective space, all triangles are "the same".

I In PG(2, q) all quadrangles are "the same".I In PG(3, q), there are two different types of quadrangles:

those contained in a plane, and those not contained in aplane.

CIRCLES IN THE PROJECTIVE PLANE

In PG(2, q), all circles, ellipses, hyperbolas, parabolas are "thesame".

PROPERTIES OF A CONIC C

(1) A line through 2 points of C has no other points of C.(2) There is a unique tangent line through each point of C.

DEFINITIONAn oval is a set of points in PG(2, q) satisfying (1) and (2).

PROPERTYAn oval contains q + 1 points.

PROPERTIES OF A CONIC C

(1) A line through 2 points of C has no other points of C.(2) There is a unique tangent line through each point of C.

DEFINITIONAn oval is a set of points in PG(2, q) satisfying (1) and (2).

PROPERTYAn oval contains q + 1 points.

OVALS IN PG(2, q)

THEOREM (SEGRE 1955)If q is odd, every oval in PG(2, q) is a conic.If q is even, there exist other examples.

SPHERES IN PG(3, q)

In PG(3, q) all elliptic quadrics are "the same".

PROPERTIES OF AN ELLIPTIC QUADRIC E

(1) A line through 2 points of E has no other points of E .(2) There is a unique tangent plane through each point of E .

DEFINITIONAn ovoid in PG(3, q) is a set of points satisfying (1)-(2).An ovoid contains q2 + 1 points.

PROPERTIES OF AN ELLIPTIC QUADRIC E

(1) A line through 2 points of E has no other points of E .(2) There is a unique tangent plane through each point of E .

DEFINITIONAn ovoid in PG(3, q) is a set of points satisfying (1)-(2).An ovoid contains q2 + 1 points.

OVOIDS IN PG(3, q)

THEOREM (BARLOTTI-PANELLA 1955)If q is odd or q = 4, every ovoid in PG(3, q) is an elliptic quadric.If q is even, there is one other family known, the Suzuki-Titsovoids.

OPEN PROBLEM

I Classification of ovoids in PG(3, q), q even.

GENERALISATION OF OVALS: ARCS

DEFINITIONAn arc is a set of points in PG(n, q), such that any n + 1 pointsgenerate the whole space.An arc in PG(2, q) is a set of points, no three of which arecollinear.

THE MAXIMUM NUMBER OF POINTS ON AN ARC

Let A be an arc in PG(2, q), then

|A| ≤ q + 2.

THE MAXIMUM NUMBER OF POINTS ON AN ARC

THEOREM (BOSE 1947)Let A be an arc in PG(2, q), q odd, then

|A| ≤ q + 1.

And if |A| = q + 1, A is a conic.

THEOREM (BOSE 1947)Let A be an arc in PG(2, q), q even, then

|A| ≤ q + 2.

ARCS AND HYPEROVALS

DEFINITIONAn arc in PG(2, q), q even, containing q + 2 points is called ahyperoval.

If q is even, all tangent lines to a conic pass through the samepoint, the nucleus.

EXAMPLEA conic and its nucleus in PG(2, q), q even, form a hyperoval.These hyperovals are the regular hyperovals.There are many other hyperovals and families of hyperovalsknown e.g. Translation, Segre, Glynn, Payne, O’Keefe,Penttila. . . hyperovals.

ARCS AND HYPEROVALS

DEFINITIONAn arc in PG(2, q), q even, containing q + 2 points is called ahyperoval.If q is even, all tangent lines to a conic pass through the samepoint, the nucleus.

EXAMPLEA conic and its nucleus in PG(2, q), q even, form a hyperoval.These hyperovals are the regular hyperovals.

There are many other hyperovals and families of hyperovalsknown e.g. Translation, Segre, Glynn, Payne, O’Keefe,Penttila. . . hyperovals.

ARCS AND HYPEROVALS

DEFINITIONAn arc in PG(2, q), q even, containing q + 2 points is called ahyperoval.If q is even, all tangent lines to a conic pass through the samepoint, the nucleus.

EXAMPLEA conic and its nucleus in PG(2, q), q even, form a hyperoval.These hyperovals are the regular hyperovals.There are many other hyperovals and families of hyperovalsknown e.g. Translation, Segre, Glynn, Payne, O’Keefe,Penttila. . . hyperovals.

OPEN PROBLEM

I Classification of hyperovals in PG(2, 2h).

GENERALISATION OF OVOIDS: CAPS

DEFINITIONA cap in PG(n, q) is a set of points, no three collinear.Note that the definitions of arcs and caps in PG(2, q) coincide.

THEOREM (BOSE 1947, QVIST 1952)Let C be a cap in PG(3, q), q even or odd, then

|C| ≤ q2 + 1.

CAPS IN PG(n, q), n > 3

If n > 3, there is no obvious classical example for a cap inPG(n, q). Only upper and lower bounds for the size of a cap inPG(n, q) are known.

OPEN PROBLEMS

I Find better lower and upper bounds for the number ofpoints on a cap in PG(n, q).

FURTHER GENERALISATION: GENERALISED OVOIDS

An ovoid is a set of q2 + 1 points in PG(3, q), no three collinear.An ovoid satisfies the property that any three points span aplane and that there is a unique tangent plane to every point ofthe ovoid.

DEFINITIONA generalised ovoid is a set of q2n + 1 (n − 1)-spaces inPG(4n − 1, q), with the property that any three elements span a(3n − 1)-space and at every element there is a unique tangent(3n − 1)-space.

OPEN PROBLEMS

I Find new examples of generalised ovals and ovoids.I Characterisation of generalised ovals and generalised

ovoids.I Classification of generalised ovals and generalised ovoids.

SPREADS OF PG(n, q)

DEFINITIONA k - spread of a projective space PG(n, q), is a set ofk -dimensional subspaces that partitions PG(n, q).

THEOREM (SEGRE 1964)There exists a k-spread of PG(n, q) ⇐⇒ (k + 1)|(n + 1).

SPREADS OF PG(n, q)

DEFINITIONA k - spread of a projective space PG(n, q), is a set ofk -dimensional subspaces that partitions PG(n, q).

THEOREM (SEGRE 1964)There exists a k-spread of PG(n, q) ⇐⇒ (k + 1)|(n + 1).

THE CONSTRUCTION OF A SPREAD

A point PG(0, pk ) of PG(n, pk )→A 1-dimensional vector space V (1, pk ) in V (n + 1, pk )→

A k -dimensional vector space V (k , p) in V (k(n + 1), p)→A (k − 1)-dimensional projective subspace PG(k − 1, p) ofPG(k(n + 1)− 1, p).

The set of points of PG(n, pk ) corresponds to a (k − 1)-spreadof PG((n + 1)k − 1, p). A spread constructed in this way iscalled a Desarguesian spread.

THE CONSTRUCTION OF A SPREAD

A point PG(0, pk ) of PG(n, pk )→A 1-dimensional vector space V (1, pk ) in V (n + 1, pk )→A k -dimensional vector space V (k , p) in V (k(n + 1), p)→

A (k − 1)-dimensional projective subspace PG(k − 1, p) ofPG(k(n + 1)− 1, p).

The set of points of PG(n, pk ) corresponds to a (k − 1)-spreadof PG((n + 1)k − 1, p). A spread constructed in this way iscalled a Desarguesian spread.

THE CONSTRUCTION OF A SPREAD

A point PG(0, pk ) of PG(n, pk )→A 1-dimensional vector space V (1, pk ) in V (n + 1, pk )→A k -dimensional vector space V (k , p) in V (k(n + 1), p)→A (k − 1)-dimensional projective subspace PG(k − 1, p) ofPG(k(n + 1)− 1, p).

The set of points of PG(n, pk ) corresponds to a (k − 1)-spreadof PG((n + 1)k − 1, p). A spread constructed in this way iscalled a Desarguesian spread.

THE CONSTRUCTION OF A SPREAD

A point PG(0, pk ) of PG(n, pk )→A 1-dimensional vector space V (1, pk ) in V (n + 1, pk )→A k -dimensional vector space V (k , p) in V (k(n + 1), p)→A (k − 1)-dimensional projective subspace PG(k − 1, p) ofPG(k(n + 1)− 1, p).

The set of points of PG(n, pk ) corresponds to a (k − 1)-spreadof PG((n + 1)k − 1, p). A spread constructed in this way iscalled a Desarguesian spread.

THE ANDRÉ-BRUCK-BOSE CONSTRUCTION

The André-Bruck-Bose construction uses a (t − 1)-spread ofPG(rt − 1, q) to construct a design.

In the case r = 2, the constructed design is a projective plane.If the spread is Desarguesian, the projective plane constructedvia A-B-B construction is Desarguesian.

SUBGEOMETRIES

If F is a subfield of K, PG(n, F) is a subgeometry of PG(n, K).Subgeometries and projections of subgeometries are oftenuseful in constructions.

If n = 2 and [K : F] = 2, then PG(2, K) is a Baer subplane ofPG(2, F).A Baer subplane is a blocking set in PG(2, K).

OPEN PROBLEMS

I Do all small minimal blocking sets arise fromsubgeometries?

I Determine the possible intersections of differentsubgeometries.

GEOMETRY AND GROUPS

THEOREMThe automorphism group of PG(V ) is induced by the group ofall non-singular semi-linear maps of V onto itself.Aut(PG(V )) acts 2-transitively on the points.

THEOREMIf Aut(Π) acts 2-transitively on the points of the projective planeΠ, then Π is Desarguesian.

AUTOMORPHISM GROUPS

Classical objects like conics, quadrics, Hermitian varieties . . . ,have classical automorphism groups:

I Quadric: orthogonal groupI Hermitian variety: unitary group

The non-classical objects have other automorphism groups:I Suzuki-Tits ovoid: Suzuki groupI Translation hyperovals: Zq × Zq−1

GEOMETRY AND GROUPS

The following questions link groups with geometry:I Given a subset S, what is Aut(S)?I Given a group G, is there a geometric object with G as its

automorphism group?