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 two dierent points there is a unique line containing them. A2 For every line l and a point P not in l, there is a unique line containing P and disjoint from l. A3 There are three points such that no line contains all three of them. Projective Plane A set, the elements of which are called points, together with a collection of subsets, called lines, satisfying the following three axioms. P1. Any two distinct points belong to exactly one line. P2. Any two distinct lines intersect in exactly one point. P3. There are four points such that no line contains any three of them.
Ane and Projective Planes Projective Closure of an Ane Plane
Ane and Projective Planes Examples
Ane and Projective Planes Examples AG(2, R) and PG(2, R)
Ane and Projective Planes Examples AG(2, R) and PG(2, R) AG(2, F p ) and PG(2, F p )
Ane and Projective Planes PG(2,2)
Ane and Projective Planes Recall... A nite projective plane of order t has t 2 + t + 1 points, t 2 + t + 1 lines, each line contains t + 1 points and and each point is contained by t + 1 lines. A nite ane plane of order t has t 2 points, t 2 + t lines, each line contains t points and and each point is contained by t + 1 lines.
Ane and Projective Planes Examples
Ane and Projective Planes Examples AG(2, R) and PG(2, R)
Ane and Projective Planes Examples AG(2, R) and PG(2, R) AG(2, F p ) and PG(2, F p )
Ane and Projective Planes Examples AG(2, R) and PG(2, R) AG(2, F p ) and PG(2, F p )...
Ane and Projective Planes Examples AG(2, R) and PG(2, R) AG(2, F p ) and PG(2, F p )... Question How can we gain further examples?
Ane and Projective Planes Idea 1 P = R R we dene the set of lines as the union of vertical lines in (i.e. x = c for all c R) and the graphs of the functions f (x) = m x 3 + b for all m, b R.
Ane and Projective Planes Idea 1 P = R R we dene the set of lines as the union of vertical lines in (i.e. x = c for all c R) and the graphs of the functions f (x) = m x 3 + b for all m, b R. WRONG! ϕ: (x, y) ( 3 x, y) is isomorphism!
Ane and Projective Planes Idea 2 P = R R We redene the set of lines in AG(2, R) as the union of vertical lines (that is, x = c for all c R) and the translates of f (x) = x 2 with all vectors v R 2.
Ane and Projective Planes Idea 2 P = R R We redene the set of lines in AG(2, R) as the union of vertical lines (that is, x = c for all c R) and the translates of f (x) = x 2 with all vectors v R 2. WRONG! Again, ϕ: (x, y) (x, x 2 + y) is isomorphism!
Ane and Projective Planes Idea 1 - WRONG!!! P = R R we dene the set of lines as the union of vertical lines in (i.e. x = c for all c R) and the graphs of the functions f (x) = m x 3 + b for all m, b R.
Ane and Projective Planes Idea 1 - WRONG!!! P = R R we dene the set of lines as the union of vertical lines in (i.e. x = c for all c R) and the graphs of the functions f (x) = m x 3 + b for all m, b R. Idea 2 - WRONG!!! P = R R We redene the set of lines in AG(2, R) as the union of vertical lines (that is, x = c for all c R) and the translates of f (x) = x 2 with all vectors v R 2.
Ane and Projective Planes Idea 1 - WRONG!!! P = R R we dene the set of lines as the union of vertical lines in (i.e. x = c for all c R) and the graphs of the functions f (x) = m x 3 + b for all m, b R. Idea 2 - WRONG!!! P = R R We redene the set of lines in AG(2, R) as the union of vertical lines (that is, x = c for all c R) and the translates of f (x) = x 2 with all vectors v R 2. Question What makes dierence? How could we deviate from PG(2, F)?
Desargues Theorem Point Perspectivity Two triangles, with their vertices named in a particular order, are said to be perspective from a point P if their three pairs of corresponding vertices are joined by concurrent lines. Figure : Point-perspective triangles with respect P
Desargues Theorem Line Perspectivity Two triangles are said to be perspective from a line l (line-perspective) if their three pairs of corresponding sides meet in collinear points. Figure : Line-perspective triangles with respect l
Desargues Theorem and Hilbert's Theorem Desargues Theorem Two triangles in R 2 are perspective from a point if and only if they are perspective from a line.
Desargues Theorem and Hilbert's Theorem Desargues Theorem Two triangles in R 2 are perspective from a point if and only if they are perspective from a line. Generalized Desargues Theorem (Hilbert) Let F be any eld or skeweld. Two triangles are perspective from a point if and only if they are perspective from a line.
Desargues Theorem and Hilbert's Theorem Desargues Theorem Two triangles in R 2 are perspective from a point if and only if they are perspective from a line. Generalized Desargues Theorem (Hilbert) Let F be any eld or skeweld. Two triangles are perspective from a point if and only if they are perspective from a line. Hilbert's Theorem A projective plane is desarguesian if and only if it can be coordinatized with a skew-eld.
Non-Desarguesian Planes Moulton Plane Figure : Moulton plane: the lines of positive slopes "refract" on the x-axis.
Non-Desarguesian Planes The Moulton Plane is not desarguesian Figure : The triangles ABC and A B C are certainly point-perspective with respect P. The lines AB,AC, B C have positive slopes and so they refract on the x-axis. It yields that AC does not meet A C on the y-axis (as it would do in AG (2, R)) and so the intersections X, Y, Z are not collinear in the Moulton plane, which disproves Desargues theorem.
Non-Desarguesian Planes Further Examples
Non-Desarguesian Planes Further Examples Shifting Planar functions such as f (x) = x 4
Non-Desarguesian Planes Further Examples Shifting Planar functions such as f (x) = x 4 Coordinatization using Nearelds
Non-Desarguesian Planes Further Examples Shifting Planar functions such as f (x) = x 4 Coordinatization using Nearelds Hall-planes
Non-Desarguesian Planes Further Examples Shifting Planar functions such as f (x) = x 4 Coordinatization using Nearelds Hall-planes...
Non-Desarguesian Planes Construction of Hall-planes
Non-Desarguesian Planes Construction of Hall-planes AG(2, q 2 ) has an obvious subplane isomorphic to AG(2, q).
Non-Desarguesian Planes Construction of Hall-planes AG(2, q 2 ) has an obvious subplane isomorphic to AG(2, q). Let D be the set of ideal points of AG(2, q) in PG(2, q 2 ).
Non-Desarguesian Planes Construction of Hall-planes AG(2, q 2 ) has an obvious subplane isomorphic to AG(2, q). Let D be the set of ideal points of AG(2, q) in PG(2, q 2 ). We redene the set of lines on AG(2, q 2 ) as the union of:
Non-Desarguesian Planes Construction of Hall-planes AG(2, q 2 ) has an obvious subplane isomorphic to AG(2, q). Let D be the set of ideal points of AG(2, q) in PG(2, q 2 ). We redene the set of lines on AG(2, q 2 ) as the union of: original lines with ideal points not in D,
Non-Desarguesian Planes Construction of Hall-planes AG(2, q 2 ) has an obvious subplane isomorphic to AG(2, q). Let D be the set of ideal points of AG(2, q) in PG(2, q 2 ). We redene the set of lines on AG(2, q 2 ) as the union of: original lines with ideal points not in D, the translates of the point-set of AG (2, q), i.e. {(a u + b, a v + c) : u, v F q } for all a, b, c F q 2, a 0.
Non-Desarguesian Planes The Hall-planes are not desarguesian Figure : Violation of the Desargues condition in Hall planes
Congurations - Arcs, Ovals and Hyperovals k-arcs A k-arc in a nite projective or ane plane is a set of k points no three of which are collinear.
Congurations - Arcs, Ovals and Hyperovals k-arcs A k-arc in a nite projective or ane plane is a set of k points no three of which are collinear. Complete arcs A k-arc is complete if it is not contained in a (k + 1)-arc.
Congurations - Arcs, Ovals and Hyperovals Bose - Theorem Theorem: an arc in a nite projective plane of order q contains at most q + 1 points if q is odd and q + 2 if q is even.
Congurations - Arcs, Ovals and Hyperovals Bose - Theorem Theorem: an arc in a nite projective plane of order q contains at most q + 1 points if q is odd and q + 2 if q is even. Ovals and Hyperovals (q + 1)-arcs and (q + 2)-arcs of a nite projective plane of order q are called ovals and hyperovals, respectively.
Congurations - Arcs, Ovals and Hyperovals Bose - Theorem Theorem: an arc in a nite projective plane of order q contains at most q + 1 points if q is odd and q + 2 if q is even. Ovals and Hyperovals (q + 1)-arcs and (q + 2)-arcs of a nite projective plane of order q are called ovals and hyperovals, respectively. Existence of Ovals and Hyperovals The existence of ovals and hyperovals in a general projective plane is a famous open question.
Congurations - Arcs, Ovals and Hyperovals Bose - Theorem Theorem: an arc in a nite projective plane of order q contains at most q + 1 points if q is odd and q + 2 if q is even. Ovals and Hyperovals (q + 1)-arcs and (q + 2)-arcs of a nite projective plane of order q are called ovals and hyperovals, respectively. Existence of Ovals and Hyperovals The existence of ovals and hyperovals in a general projective plane is a famous open question. Note Choosing the elements greedily one can generate a complete - arc of size at least q.
Ovals and Hyperovals in PG(2, q) Theorem There exist ovals and hyperovals in PG(2, q).
Ovals and Hyperovals in PG(2, q) Theorem There exist ovals and hyperovals in PG(2, q). Proof The graph of f (x) = x 2 together with the ideal point (0, 1, 0) form an oval. If q = 2 k, this oval can be completed to a hyperoval by adding point (1, 0, 0).
Ovals and Hyperovals in PG(2, q) Theorem There exist ovals and hyperovals in PG(2, q). Proof The graph of f (x) = x 2 together with the ideal point (0, 1, 0) form an oval. If q = 2 k, this oval can be completed to a hyperoval by adding point (1, 0, 0). Note In a nite projective plane of even order ovals can be completed to hyperovals uniquely.
Complete Arcs in PG(2, q) Theorem If K is a k-arc in PG(2, q) and 1 k q+4, if q is even, 2 2 k 2q+5 3, if q is odd, then K is contained in a unique complete arc. Theorem (Segre) If K is a k-arc in PG(2, q) and 1 k > q q + 1, if q is even, 2 k > q q 4 + 7 4, if q is odd, then K can be completed to an oval. Corollary No complete arc of size q exists in PG(2, q).
Complete Arcs in Hall-Planes Theorem (Sz nyi) The set of points C dened by the function f (x) = x 2 (x F q 2\F q ) together with (0, 1, 0) form a complete q 2 q + 1 arc in Hall(q 2 ) for odd q. Theorem (Sz nyi) For an arbitrary, xed non-square c F q 2 the hyperbola H = {(x, c 0 F 2)} is a complete x q (q2 1)-arc in Hall(q 2 ).
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