an introduction to finite geometry - ghent...

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.

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?

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

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.



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.



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.

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.

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).

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.


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).



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 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?

an introduction to finite geometry - ghent...

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