An introduction to Finite Geometry

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1 An introduction to Finite Geometry Geertrui Van de Voorde Ghent University, Belgium Pre-ICM International Convention on Mathematical Sciences Delhi

2 INCIDENCE STRUCTURES EXAMPLES Designs Graphs Linear spaces Polar spaces Generalised polygons Projective spaces... Points, vertices, lines, blocks, edges, planes, hyperplanes... + incidence relation

3 PROJECTIVE SPACES Many examples are embeddable in a projective space. V : Vector space PG(V ): Corresponding projective space

4 FROM VECTOR SPACE TO PROJECTIVE SPACE

5 FROM VECTOR SPACE TO PROJECTIVE SPACE The projective dimension of a projective space is the dimension of the corresponding vector space minus 1

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

7 PROPERTIES OF A PG(V ) OF DIMENSION 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).

8 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 ).

9 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.

10 PROJECTIVE PLANES Points, lines and three axioms (a) r s!l (b) L M!r (c) r, s, t, u If Π is a projective plane, then interchanging points and lines, we obtain the dual plane Π D.

11 FINITE PROJECTIVE PLANES DEFINITION The order of a projective plane is the number of points on a line minus 1. A projective plane of order n has n 2 + n + 1 points and n 2 + n + 1 lines.

12 PROJECTIVE SPACES OVER A FINITE FIELD F p = Z/Z p if p is prime F q = F p [X]/(f (X)), with f (X) an irreducible polynomial of degree h if q = p h, p prime. NOTATION V (F d q) = V (d, F q ) = V (d, q): vector space in d dimensions over F q. The corresponding projective space is denoted by PG(d 1, q).

13 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.

14 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.

15 WHEN IS A PROJECTIVE PLANE = PG(2, q)? THEOREM A finite projective plane = PG(2, q) Desargues configuration holds for any two triangles that are in perspective.

16 DESARGUES CONFIGURATION

17 EXISTENCE AND UNIQUENESS OF A PROJECTIVE PLANE OF ORDER n PG(2, q) is an example of a projective plane of order q = p h, p prime. Is this the only example of a projective plane of order q = p h?

18 EXISTENCE AND UNIQUENESS OF A PROJECTIVE PLANE OF ORDER n PG(2, q) is an example of a projective plane of order q = p h, p prime. Is this the only example of a projective plane of order q = p h? Are there projective planes of order n, where n is not a prime power?

19 THE SMALLEST PROJECTIVE PLANE: PG(2, 2) The projective plane of order 2, the Fano plane, has: q + 1 = = 3 points on a line, 3 lines through a point. And it is unique.

20 THE PROJECTIVE PLANE PG(2, 3) The projective plane PG(2, 3) has: q + 1 = = 4 points on a line, 4 lines through a point. And it is unique.

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

22 SMALL PROJECTIVE PLANES 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.

23 SMALL PROJECTIVE PLANES 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 (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?

24 SMALL PROJECTIVE PLANES 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 (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, BY COMPUTER) There is no projective plane of order 10

25 OPEN QUESTIONS Do there exist projective planes with the order not a prime power? How many non-isomorphic projective planes are there of a certain order?

26 FIRST GEOMETRICAL OBJECTS: SUBSETS

27 TRIANGLES AND QUADRANGLES IN PROJECTIVE SPACE In a projective space, all triangles are "the same".

28 TRIANGLES AND QUADRANGLES IN PROJECTIVE SPACE In a projective space, all triangles are "the same". In PG(2, q) all quadrangles are "the same".

29 TRIANGLES AND QUADRANGLES IN PROJECTIVE SPACE In a projective space, all triangles are "the same". In PG(2, q) all quadrangles are "the same". In PG(3, q), there are two different types of quadrangles: those contained in a plane, and those not contained in a plane.

30 CIRCLES IN THE PROJECTIVE PLANE In PG(2, q), all circles, ellipses, hyperbolas, parabolas are "the same".

31 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.

32 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.

33 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.

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

35 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.

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

37 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-Tits ovoids.

38 OPEN PROBLEM Classification of ovoids in PG(3, q), q even.

39 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.

40 THE MAXIMUM NUMBER OF POINTS ON AN ARC Let A be an arc in PG(2, q), then A q + 2.

41 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.

42 ARCS AND HYPEROVALS DEFINITION An arc in PG(2, q), q even, containing q + 2 points is called a hyperoval.

43 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.

44 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.

45 OPEN PROBLEM Classification of hyperovals in PG(2, 2 h ).

46 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 q

47 CAPS IN 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.

48 OPEN PROBLEMS Find better lower and upper bounds for the number of points on a cap in PG(n, q).

49 FURTHER GENERALISATION: GENERALISED OVOIDS An ovoid is a set of q 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 q 2n + 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.

50 OPEN PROBLEMS Find new examples of generalised ovals and ovoids. Characterisation of generalised ovals and generalised ovoids. Classification of generalised ovals and generalised ovoids.

51 SPREADS OF 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).

52 SPREADS OF 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).

53 THE CONSTRUCTION OF A SPREAD A point PG(0, p k ) of PG(n, p k ) A 1-dimensional vector space V (1, p k ) in V (n + 1, p k )

54 THE CONSTRUCTION OF A SPREAD A point PG(0, p k ) of PG(n, p k ) A 1-dimensional vector space V (1, p k ) in V (n + 1, p k ) A k-dimensional vector space V (k, p) in V (k(n + 1), p)

55 THE CONSTRUCTION OF A SPREAD A point PG(0, p k ) of PG(n, p k ) A 1-dimensional vector space V (1, p k ) in V (n + 1, p k ) 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).

56 THE CONSTRUCTION OF A SPREAD A point PG(0, p k ) of PG(n, p k ) A 1-dimensional vector space V (1, p k ) in V (n + 1, p k ) 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, p k ) corresponds to a (k 1)-spread of PG((n + 1)k 1, p). A spread constructed in this way is called a Desarguesian spread.

57 THE ANDRÉ-BRUCK-BOSE CONSTRUCTION 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.

58 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).

59 OPEN PROBLEMS Do all small minimal blocking sets arise from subgeometries? Determine the possible intersections of different subgeometries.

60 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.

61 AUTOMORPHISM GROUPS Classical objects like conics, quadrics, Hermitian varieties..., have classical automorphism groups: Quadric: orthogonal group Hermitian variety: unitary group The non-classical objects have other automorphism groups: Suzuki-Tits ovoid: Suzuki group Translation hyperovals: Z q Z q 1

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

Congurations in Non-Desarguesian Planes

Congurations in Non-Desarguesian Planes CEU Budapest, Hungary September 25, 2012 Ane and Projective Planes Ane Plane A set, the elements of which are called points, together with a collection of subsets, called lines, satisfying A1 For every