Basic concepts

Definition 1.1   If $W \subset {\Bbb R}^{n+1}$ is a vector subspace, the set of all lines in $W$ passing through the origin is a projective subspace of ${\Bbb R}P^n$ that we shall denote by $P(W)$. If the dimension of $W$ equals $k+1$, we say that $P(W)$ is a $k$-dimensional projective subspace. A projective subspace of dimension one is called a projective line and a projective subspace of codimension one is called a projective hyperplane.

Exercise 1.1 (00)   Show that projective transformations of ${\Bbb R}P^n$ send $k$-dimensional projective subspaces to $k$-dimensional projective subspaces.

Exercise 1.2 (* 05)   Let $V$ be a four-dimensional vector space over a field ${\Bbb F}$ with $p$ elements. How many projective lines are then in $P(V)$? How many projective hyperplanes?

Exercise 1.3 (10)   Verify that projective lines in ${\Bbb R}P^2$ have the following two properties:

• Any two projective lines intersect at one point.
• Any two distinct points are joined by a unique projective line.

Figure 1 shows that the set of lines in ${\Bbb R}^3$ that pass through the origin and are not horizontal can be identified with the set of points on the plane $z = 1$. Note also how a straight line on the plane corresponds to a projective line on ${\Bbb R}P^2$.

Figure 1.

Exercise 1.4 (00)   Show that ${\Bbb R}P^2$ minus a projective line can be identified with ${\Bbb R}^2$.

The analytic version of the figure above is as follows:

Exercise 1.5 (00)   Let $U_{z}$ be the set of points in the projective plane whose homogeneous coordinates $[x:y:z]$ have $z \neq 0$ (these correspond to lines in ${\Bbb R}^3$ that pass through the origin and are not horizontal). Show that the map $\varphi_{z} : U_{z} \rightarrow {\Bbb R}^2$ given by $[x:y:z] \mapsto (x/z,y/z)$ is a bijection.

Exercise 1.6 (10)   Consider now ${\Bbb R}P^n$ and define $U_{i} \subset {\Bbb R}P^n$ to be the set of points whose homogeneous coordinates $[x_{1}: \dots :x_{n+1}]$ have $x_{i} \neq 0$. Show that

1. The union of the $U_{i}, 1 \leq i \leq n+1$, is all of ${\Bbb R}P^n$.
2. The map $\varphi_{i} : U_{i} \rightarrow {\Bbb R}^n$ given by
   $$[x_{1}: \dots : x_{n+1}] \mapsto (x_{1}/x_{i}, \dots, x_{i-1}/x_{i}, x_{i+1}/x_{i}, \dots, x_{n+1}/x_{i})$$
   is a bijection.
3. Compute the map $\varphi_{i} \circ \varphi_{j}^{-1}$ defined on $\varphi_{j}(U_{i} \cap U_{j}) \subset {\Bbb R}^n$.

The maps $\varphi_{i} : U_{i} \rightarrow {\Bbb R}^n$ are called charts and they can be used to work on projective space as if it were ${\Bbb R}^n$. When we do this, the projective hyperplane given by all points whose homogenous coordinates are of the form $[x_{1}: \dots: x_{i-1}:0:x_{i+1}:\dots:x_{n+1}]$ is called the hyperplane at infinity (visualize this in figure 1). These are exactly the points that are not in $U_{i}$ and are not taken into account by our representation. As the next exercise shows, we can make any projective hyperplane be our hyperplane at infinity.

Exercise 1.7 (05)   Deduce from the previous exercise that if $P(W) \subset {\Bbb R}P^n$ is a projective hyperplane, then ${\Bbb R}P^n \setminus P(W)$ can be identified with ${\Bbb R}^n$. Give a picture proof in the case $n = 2$.

We now describe the action of $PGL(n+1;{\Bbb R})$ using the charts of ${\Bbb R}P^n$. To keep things simple, we only consider the chart $\varphi_{n+1}$.

If $A$ is an invertible $(n+1) \times (n+1)$ matrix, then its action on ${\Bbb R}P^n$ in homogeneous coordinates is given simply by $[v] \mapsto [Av]$. If $(y_{1}, \dots, y_{n}) \in {\Bbb R}^n$, we identify it with the point $[y_{1}:\dots :y_{n}:1] \in {\Bbb R}P^n$. The matrix $A$ sends this point to the point

$$\left[\sum_{i = 1}^n a_{1,i} y_{i} + a_{1,n+1}: \dots : \sum_{i = 1}^n a_{n+1, i} y_{i} + a_{n+1,n+1} \right] \in {\Bbb R}P^n .$$

If we use $\varphi$ to identify this point with a point in ${\Bbb R}^n$, we obtain $(x_{1}, \dots x_{n})$, where

$$x_{i} := \frac{a_{i,1}y_{1} + \cdots + a_{i,n}y_{n} + a_{i, n+1}}{ a_{n+1,1}y_{1} + \cdots + a_{n+1,n}y_{n} + a_{n+1,n+1}} .$$

If it happens that the denominator equals zero, we say that the point $(y_{1},\dots,y_{n})$ is sent to infinity.