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 INCIDENCE STRUCTURES 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 ) 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 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. 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 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. 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. DESARGUES CONFIGURATION I Are there projective planes of order n, where n is not a prime 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, p prime. I Is this the only example of a projective plane of order q = ph? EXISTENCE AND UNIQUENESS OF A PROJECTIVE PLANE OF ORDER 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? 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. 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 SMALL PROJECTIVE PLANES 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, 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. THEOREM There are 4 non-isomorphic planes of order 9. 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. 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? 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 OPEN QUESTIONS 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? FIRST GEOMETRICAL OBJECTS: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. TRIANGLES AND QUADRANGLES IN PROJECTIVE SPACE 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. 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". 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 a plane. CIRCLES IN THE PROJECTIVE PLANE 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. SPHERES IN 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. 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. 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 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-Tits ovoids. OPEN PROBLEM 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.