lines through a point of the projective plane

Question

I'm having difficulty understanding a particular example of Mumford's "Red Book". In exemple D, first chapter, he considers the set of lines passing through a point of $\mathbb{P}_2$ (do we call it pencil?), and says that we identify it with $\mathbb{P}_1$, in a classical way. It might be a stupid question, or just me missing something but I would like to understand this fact and what does he mean by classical way. Thank you

Answer 1

a) Indeed there is a very classical duality between points and lines in projective space $\mathbb P^2$.
Points $P=(x:y:z)\in \mathbb P^2$ live in one projective plane and lines $l\subset \mathbb P^2$, which are subsets of that plane, can also been seen as points in a new projective space $ { \mathbb P^2}^*$.
To the line $l\subset \mathbb P^2$ with equation $ax+by+cz=0$ one just associates the point $ [l]=(a:b:c)\in { \mathbb P^2}^*$ .

b) To get back to your question: if you fix a point $P_0=(x_0:y_0:z_0)\in \mathbb P^2$, a line $l$ given by $ax+by+cz=0 $ will go through $P_0$ if and only if $ax_0+by_0+cz_0=0$.
In the dual perspective the set of all the corresponding points $[l]\in { \mathbb P^2}^*$ will describe a line in the new plane: $L_{P_0}\subset { \mathbb P^2}^*$ called, as you correctly stated, a pencil of lines.
The equation of $L_{P_0}$ is $x_0a+y_0b+z_0c=0$, where now $a, b $ and $c$ are seen as variables!
Concretely: if $P_0=(1:2:-1)$, the pencil $L_{P_0}$ consists of the lines $[l]\in { \mathbb P^2}^*$whose equation $ax+by+cz=0$ satisfies $a+2b-c=0$.
So the line $3x+4y+11z=0$ belongs to the pencil but the line $x-2y+4z=0$ does not.

c) This is the beginning of the theory of duality of projective spaces, an intensively studied subject in algebraic geometry, created at the beginning of the 19th century by two officers in the French army, Gergonne and Poncelet, who quarreled bitterly over the priority of this discovery...

Answer 2

Yes, a family of lines passing through a common point is called a pencil of lines.

One way to view this identification is the following: the projective line $\mathbb P^1$ can be written using homogeneous coordinates, where each point corresponds to an equivalence class of coordinates. All vectors of an equivalence class form a line, omitting the origin but that's a minor point. So you might as well say that a point in $\mathbb P^1$ corresponds to a line through the origin (a linear subspace) in $\mathbb R^2$. But you find that same structure in $\mathbb P^2$ as well. For the family of lines through a given point, it doesn't matter whether you include points at infinity or not. They are isomorphic to the projective line.

Another definition would go via cross ratios. If you fix three points on a projective line, all other points are uniquely determined with respect to these three by their cross ratio. Likewise, if you fix three concurrent lines, then any other line through their common point is uniquely determined by its cross ratio to these three. You can define the cross ratio of four lines as the cross ratios of the points of intersection they form with any other line not passing through their common point.