I know what real projective line is. But what is a formal definition of a projective line? Unfortunately there is nothing on Wikipedia about it. They are always talking about projective line as a part of projective plane. How would one describe projective line without any underlying space?
Thanks.
You can think of the Points of the real projective plane as the lines through the origin in $\mathbb{R}^3$. The Lines in that projective space are the real planes through the origin. Two Points in the projective space are joined by the Line that is the plane they span.
So the Points of the projective line are the lines through the origin in $\mathbb{R}^2$.
Alternatively, you can think of the projective line as an ordinary Euclidean line with a single extra point "at infinity". In that construction the geometry on the projective line is determined by the cross ratio.
A third view is that the projective line is the set of points on a circle with diagonally oppposite points identified.
Hartshorne's little book on Projective Geometry (Foundations of Projective Geometry) has a wonderful intro to the subject. He first describes an Affine Plane, which consists of a set $Q$, called the points of the geometry, and a collection $R$ of subsets of $Q$ called the lines of the geometry. Note that the set $Q$ could be a set of numbers, functions, cheese omelets ... whatever. In order for $Q$ and $R$ to be an affine plane, they must have three properties:
A1. If $P,T \in Q$ and $P \ne T$, then there is a unique line $\ell \in R$ with $P \in \ell$ and $Q \in \ell$. (Short form: through any two distinct points, there's a unique line.)
A2. Given a point $P$ and a line $\ell \in R$, with $P \notin \ell$, there is a unique line $m \in R$ such that (i) $P \in m$, and (ii) $\ell \cap m = \emptyset$. (Short form: Through any point $P$ not on $\ell$, there's a line $m$ parallel to $\ell$.)
Note: Hartshorne defines "parallel" thus: $\ell$ and $m$ are parallel if either (a) $\ell = m$ or (b) $\ell \cap m = \emptyset$.
A3: There are points $A, B, C \in P$ that are non-collinear, i.e., such that there is no line $\ell \in R$ that contains all three.
That's a very abstract definition of an affine plane, but you can prove some interesting things already. One is that every affine plane contains at least four points. Another is that there is a 4-point affine plane.
One thing that's a little disappointing for folks who love symmetry is that for every pair of distinct points, there's a line containing them, but for some pairs of distinct lines, there's no point that's on both (this happens for lines $\ell$ and $m$ that a parallel and distinct). Hartshorne then describes a projective plane, which actually has this kind of symmetry, by saying once again it's a set $A$ of things called points, and a set $B$ of subsets of $A$ having four properties:
P1: two distinct points lie on exactly one line.
P2: any two lines meet in at least one point
P3: There exist three noncollinear points
P4: Every line contains at least three points.
He further shows how to take any affine plane and create a projective plane by adding a few new points and lines, calling this the projectivization of the affine plane. (He also shows that if you take a projective plane and delete all the points of one single line $\ell$ (deleting those points from other lines as well), you end up with an affine plane. So there's a way (projectivization) to go from affine planes to projective ones, and a way (line-deletion) to go the other direction. If we do both operations, we have a way to take an affine plane and produce an affine plane. Do we get the same one back again? Similarly, if we delete a line from a projective plane and then projectivize the result, do we get the same projective plane back again? Hartshorne raises these questions, and then takes several chapters to finally resolve them in a somewhat surprising way.
So...what's a projective line? In Hartshorne's book, it's one of the selected subsets that are named "lines" in the definition of a projective plane.
I know that's not very satisfying, but it's how it's done in "synthetic projective geometry", at least. (Or at least in a slightly simplified version of synthetic projective geometry.)
@EthanBolker has described for you what is a projective line in so-called "real projective geometry". Hartshorne's book also discusses this, and shows that the real projective plane (in any of its guises) is an instance of the thing he's called a "projective plane." He then shows that if you add a few more requirements to the abstract definition of a projective plane, you can show that it's actually the projective plane for some division ring, i.e., it's very nearly like the real projective plane, except that the coordinates, rather than being real numbers, are elements of some division ring, but otherwise all the algebra you can do on the real projective plane also works in these.
