Projective straight line

projective line

A projective space of dimension $ 1 $. A projective line, considered as an independent object, is a closed one-dimensional manifold. A projective line is a special (and peculiar) projective space: there are no interesting incidence relations on it similar to those on projective spaces of higher dimensions. The only invariant of a projective line is the number of its points. A projective line is called continuous, discrete or finite if it is incident to a point set of the cardinality of the continuum, of countable cardinality or of finite cardinality, respectively.

A projective line is called ordered if a relation of separation of two pairs of distinct points is given on it. It is assumed that this relation does not depend on the order of the pairs or on the order of the points in the pairs, and that any quadruple of distinct points splits uniquely into two mutually separating pairs. In addition, the location axiom is adopted, which connects five distinct points (see, for example, [1]). The ordering of a projective line over the field $ \mathbf R $ is related to the ordering of the field $ \mathbf R $. Namely, a pair of points $ \{ A , B \} $ separates a pair $ \{ C , D \} $ if the cross ratio $ ( A , B ; C , D ) $ is negative, and does not separate if $ ( A , B ; C , D ) $ is positive. The finite projective line $ \mathop{\rm PG} ( 1, q ) $ over a Galois field of odd order $ q $ can be ordered similarly to the real projective line. It is assumed (see [4]) that a pair $ \{ A , B \} $ of points separates a pair $ \{ C , D \} $ if and only if $ ( A , B ; C , D ) $ is a quadratic residue in the Galois field $ \mathop{\rm GF} ( q ) $.

A projective line acquires a certain geometric structure if it is imbedded in a projective space of higher dimension. For example, a projective line is uniquely determined by two distinct points, while the analytic definition of a projective line as a set of equivalence classes of pairs of elements of a skew-field $ k $ that are not simultaneously equal to zero, is actually equivalent to an imbedding of the projective line into a projective space $ P _ {n} ( k ) $, $ n \geq 2 $. If $ P _ {1} ( k ) $ is the projective line over the field $ k $, then the group $ \mathop{\rm Aut} P _ {1} ( k ) $ of automorphisms of the projective line can be represented on the points of $ P _ {1} ( k ) $ in parametric form as the set of mappings

$$ k \rightarrow \frac{k ^ \alpha a + b }{k ^ \alpha c + d } ,\ a , b ,\ c , d \in k ,\ ad - bc \neq 0 ,\ \alpha \in \mathop{\rm Aut} k . $$

The group of algebraic automorphisms of the real projective line is isomorphic to the group of displacements of the real hyperbolic plane, and the order of the group $ \mathop{\rm Aut} \mathop{\rm PG} ( 1 , p ^ {h} ) $ equals $ h ( p ^ {3h} - p ^ {h} ) $.

On a projective line one can construct different geometries. For example, the Möbius plane of order $ p $ admits an interpretation on the projective line $ \mathop{\rm PG} ( 1, p ^ {2} ) $( see [5]). Another traditional geometric construction is the representation of the projective space $ P _ {n} ( k ) $ on the projective line $ P _ {1} ( k ) $( see [2]) under which a point of $ P _ {n} ( k ) $ is represented by an $ n $- tuple of points of the projective line $ P _ {1} ( k ) $( here $ k $ is an algebraically closed field).

References

[1]	N.A. Glagolev, "Projective geometry" , Moscow (1963) (In Russian)
[2]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[3]	D.R. Hughes, F.C. Piper, "Projective planes" , Springer (1973) MR0333959 Zbl 0267.50018
[4]	P. Kustaanheimo, "On a relation of order in geometries over a Galois field" Soc. Sci. Fenn. Comment. Phys.-Math. , 20 : 8 (1957) MR100245 Zbl 0088.13103
[5]	O. Veblen, J.W. Young, "Projective geometry" , 1 , Blaisdell (1938) MR0179667 MR0179666 MR1519256 MR1506049 MR1500790 MR1500747 Zbl 0127.37604 Zbl 0018.32604 Zbl 63.0693.02 Zbl 55.0413.02 Zbl 52.0732.01 Zbl 51.0591.05 Zbl 51.0569.04 Zbl 49.0547.01 Zbl 48.0843.04 Zbl 47.0582.08 Zbl 41.0606.06 Zbl 39.0606.01 Zbl 38.0562.01

Comments

The relation of separation (for a projective line over $ \mathbf R $) is invariant under projective transformations, as follows from the above-mentioned properties of cross ratios.

If all classical separation axioms hold, then the coordinate field $ k $ has characteristic zero, and is infinite.

References

[a1]	J.W.P. Hirschfeld, "Projective geometries over finite fields" , Clarendon Press (1979) MR0554919 Zbl 0418.51002