An Introduction to Finite Geometry

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An introduction to Finite Geometry

Geertrui Van de Voorde

Ghent University, Belgium

Pre-ICM International Convention on Mathematical Sciences Delhi INCIDENCESTRUCTURES

EXAMPLES

I Designs

I Graphs

I Linear spaces

I Polar spaces

I Generalised polygons

I Projective spaces

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

Many examples are embeddable in a projective space.

V : Vector space PG(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 dimension of the corresponding vector space minus 1 PROPERTIES OF A PG(V ) OFDIMENSION d

(1) Through every two points, there is exactly one line. PROPERTIES OF A PG(V ) OFDIMENSION d

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

(3) There are d + 2 points such that no d + 1 of them are contained in a (d − 1)-dimensional projective space PG(d − 1, q). DEFINITION If d = 2, a space satisfying (1)-(2)-(3) is called a projective plane.

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

THEOREM (VEBLEN-YOUNG 1916) If d ≥ 3, a space satisfying (1)-(2)-(3) is a d-dimensional PG(V ). WHICH SPACES SATISFY (1)-(2)-(3)?

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

DEFINITION If d = 2, a space satisfying (1)-(2)-(3) is called a projective plane. PROJECTIVEPLANES

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. FINITEPROJECTIVEPLANES

DEFINITION The order of a projective plane is the number of points on a line minus 1.

A projective plane of order n has n2 + n + 1 points and n2 + n + 1 lines. PROJECTIVE SPACES OVER A FINITE FIELD

Fp = Z/Zp if p is prime Fq = Fp[X]/(f (X)), with f (X) an irreducible polynomial of degree h if q = ph, p prime. NOTATION d V (Fq ) = V (d, Fq) = V (d, q): vector space in d dimensions over Fq. The corresponding projective space is denoted by PG(d − 1, q). PG(2, q) is not the only example of a projective plane, there are other 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, and there are q + 1 lines through a point. PROJECTIVE PLANES OVER A FINITE FIELD

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

THEOREM A ﬁnite projective plane =∼ PG(2, q) ⇐⇒ Desargues conﬁguration holds for any two triangles that are in perspective. DESARGUES CONFIGURATION I Are there projective planes of order n, where n is not a prime power?

EXISTENCEANDUNIQUENESSOFAPROJECTIVEPLANE OFORDER n

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

I Is this the only example of a projective plane of order q = ph? EXISTENCEANDUNIQUENESSOFAPROJECTIVEPLANE OFORDER n

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

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

I Are there projective planes of order n, where n is not a prime power? THESMALLESTPROJECTIVEPLANE: 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. THEPROJECTIVEPLANE 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. THEOREM There 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 2 mod 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, BYCOMPUTER) There is no projective plane of order 10

SMALLPROJECTIVEPLANES

The projective planes PG(2, 4), PG(2, 5), PG(2, 7) and PG(2, 8) are unique. THEOREM (BRUCK-CHOWLA-RYSER 1949) Let n be the order of a projective plane, where n =∼ 1 or 2 mod 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, BYCOMPUTER) There is no projective plane of order 10

SMALLPROJECTIVEPLANES

The projective planes PG(2, 4), PG(2, 5), PG(2, 7) and PG(2, 8) are unique. THEOREM There are 4 non-isomorphic planes of order 9. THEOREM (LAM,SWIERCZ,THIEL, BYCOMPUTER) There is no projective plane of order 10

SMALLPROJECTIVEPLANES

The projective planes PG(2, 4), PG(2, 5), PG(2, 7) and PG(2, 8) are unique. THEOREM There are 4 non-isomorphic planes of order 9.

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

I How many non-isomorphic projective planes are there of a certain order? FIRSTGEOMETRICALOBJECTS:SUBSETS 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 a plane.

TRIANGLESANDQUADRANGLESINPROJECTIVESPACE

I In a projective space, all triangles 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 a plane.

TRIANGLESANDQUADRANGLESINPROJECTIVESPACE

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

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

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 a plane. CIRCLESINTHEPROJECTIVEPLANE

In PG(2, q), all circles, ellipses, hyperbolas, parabolas are "the same". DEFINITION An oval is a set of points in PG(2, q) satisfying (1) and (2).

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

PROPERTY An 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. SPHERESIN PG(3, q)

In PG(3, q) all elliptic quadrics are "the same". DEFINITION An ovoid in PG(3, q) is a set of points satisfying (1)-(2). An ovoid contains q2 + 1 points.

PROPERTIESOFANELLIPTICQUADRIC 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. PROPERTIESOFANELLIPTICQUADRIC 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. DEFINITION An ovoid in PG(3, q) is a set of points satisfying (1)-(2). An ovoid contains q2 + 1 points. OVOIDSIN 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-Tits ovoids. OPENPROBLEM

I Classiﬁcation of ovoids in PG(3, q), q even. GENERALISATION OF OVALS: ARCS

DEFINITION An arc is a set of points in PG(n, q), such that any n + 1 points generate the whole space. An arc in PG(2, q) is a set of points, no three of which are collinear. THEMAXIMUMNUMBEROFPOINTSONANARC

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

|A| ≤ q + 2. THEMAXIMUMNUMBEROFPOINTSONANARC

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. If q is even, all tangent lines to a conic pass through the same point, the nucleus. EXAMPLE A 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 hyperovals known e.g. Translation, Segre, Glynn, Payne, O’Keefe, Penttila. . . hyperovals.

ARCS AND HYPEROVALS

DEFINITION An arc in PG(2, q), q even, containing q + 2 points is called a hyperoval. There are many other hyperovals and families of hyperovals known e.g. Translation, Segre, Glynn, Payne, O’Keefe, Penttila. . . hyperovals.

ARCS AND HYPEROVALS

DEFINITION An arc in PG(2, q), q even, containing q + 2 points is called a hyperoval. If q is even, all tangent lines to a conic pass through the same point, the nucleus. EXAMPLE A 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 hyperovals known e.g. Translation, Segre, Glynn, Payne, O’Keefe, Penttila. . . hyperovals. OPENPROBLEM

h I Classiﬁcation of hyperovals in PG(2, 2 ). GENERALISATION OF OVOIDS: CAPS

DEFINITION A 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. CAPSIN PG(n, q), n > 3

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

I Find better lower and upper bounds for the number of points on a cap in PG(n, q). FURTHER GENERALISATION: GENERALISEDOVOIDS

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 a plane and that there is a unique tangent plane to every point of the ovoid. DEFINITION A generalised ovoid is a set of q2n + 1 (n − 1)-spaces in PG(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. OPENPROBLEMS

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

SPREADSOF PG(n, q)

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

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

THEOREM (SEGRE 1964) There exists a k-spread of PG(n, q) ⇐⇒ (k + 1)|(n + 1). A k-dimensional vector space V (k, p) in V (k(n + 1), p)→ A (k − 1)-dimensional projective subspace PG(k − 1, p) of PG(k(n + 1) − 1, p).

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

THECONSTRUCTIONOFASPREAD

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

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

THECONSTRUCTIONOFASPREAD

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

THECONSTRUCTIONOFASPREAD

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) of PG(k(n + 1) − 1, p). THECONSTRUCTIONOFASPREAD

The set of points of PG(n, pk ) corresponds to a (k − 1)-spread of PG((n + 1)k − 1, p). A spread constructed in this way is called a Desarguesian spread. THE ANDRÉ-BRUCK-BOSECONSTRUCTION

The André-Bruck-Bose construction uses a (t − 1)-spread of PG(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 constructed via 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 often useful in constructions.

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

I Do all small minimal blocking sets arise from subgeometries?

I Determine the possible intersections of different subgeometries. GEOMETRY AND GROUPS

THEOREM The automorphism group of PG(V ) is induced by the group of all non-singular semi-linear maps of V onto itself. Aut(PG(V )) acts 2-transitively on the points. THEOREM If Aut(Π) acts 2-transitively on the points of the projective plane Π, then Π is Desarguesian. AUTOMORPHISMGROUPS

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

I Quadric: orthogonal group

I Hermitian variety: unitary group

The non-classical objects have other automorphism groups:

I Suzuki-Tits ovoid: Suzuki group I 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?