Lecture 1: Introduction to Projective Geometry

Let $V$ be a finite dimensional vector space over a field $\mathbb{F}$.

The projectiviziation of V is \[\mathbb{P}(V) = (V\backslash {0})/\mathbb{F}^\times = (V\backslash{0})/\sim\] where we say $v \sim w$ if $w = \lambda v$ for some nonzero $\lambda \in \mathbb{F}$.

If $V = \mathbb{F}^{n+1}$ then we write $\mathbb{P}(\mathbb{F}^{n+1}) = \mathbb{F} P^n$.

One way to understand the projectivization of $V$ is as the space of 1-dimensional subspaces of $V$.

If we take $\mathbb{F}=\mathbb{R}$ or $\mathbb{F} = \mathbb{C}$, then we can consider $\mathbb{F} P^1$ to be the field $\mathbb{F}$ together with a point at infinity via the following map:

A transformation $T \in GL(V)$ sends 1-dimensional subspaces to 1-dimensional subspaces. Thus $T$ induces a map $[T]:\mathbb{P}(V) \to \mathbb{P}(V)$, which we call a projective transformation.

We now consider the case of $\mathbb{F} P^1$ further. Suppose $T\in GL(\mathbb{F}^2)$, and write $T = \begin{pmatrix} a&b \\ c&d \end{pmatrix}$. Then \[[T]\left[\left(\begin{array}{c} x_0, \ x_1 \ \end{array}\right)\right] = \left[\left(\begin{array}{c} ax_0 + bx_1, \ cx_0 + dx_1 \ \end{array}\right)\right]. \]

Applying the map $\theta$ above, we find $\theta([(x_0,x_1)]) = x_0/x_1 = y$ and that $\theta([T][(x_0,x_1)]) = (ay+b)/(cy+d)$. We conclude that the group of projective transformations is isomorphic to the Mobius transformations.

We now note that $T \in GL(V)$ acts trivially on $\mathbb{P}(V)$ if and only if $T$ fixes every 1-dimensional subspace of $V$, which occurs if and only if $T = \lambda \cdot \text{Id}$ for some nonzero $\lambda$.

Definition: The projective general linear group is \[PGL(V) = GL(V)/{\lambda \cdot \text{Id} : \lambda \neq 0}.\]

Exercise: The set of all nonzero multiples of the identity is the center of $GL(V)$.

Definition: If $U \neq 0$ is a vector subspace of $V$, then $\mathbb{P}(U)$ is a projective subspace of $\mathbb{P}(V)$.

• If $\dim U = 1$ then $\mathbb{P}(U)$ is one point.
• If $\dim U = 2$ then $\mathbb{P}(U)$ is a projective line.
• If $\dim U = \dim V - 1$ then $\mathbb{P}(U)$ is a projective hyperplane.

The following definition is necessary to define a projective basis.

Definition: A set of points $S=\{p_i : I\in I\}$ in $\mathbb{P}(V)$ is in general position if for all $k \leq \dim(V)$, every subset of $k$ points in $S$ is not contained in a projective subspace of dimension $k-2$.

So for example, any subset of 3 points of $S$ must not lie on a projective line.

A projective basis of $\mathbb{F} P^n$ is a set of $n+2$ points in general position.

Theorem: There exists a unique projective transformation taking one ordered projective basis to another.

Proof: Fix a basis $v_1,\ldots,v_n$ of $V$. Then $[v_1],[v_2],\ldots,[v_n],[v_1 + \cdots + v_n] = [v_{n+1}]$ is a projective basis for $\mathbb{P}(V)$. Given another projective basis $[w_1],\ldots,[w_{n+1}]$, it will suffice to show that there is a unique projective transformation taking this basis to our chosen one.

Now, since $w_1,\ldots,w_n$ is a basis for $V$, there exists a vector space isomorphism $T$ sending $v_i$ to $w_i$ for all $1\le i \le n$. Note that $[T]$ is not the unique transformation taking $[v_i]$ to $[w_i]$ because of scaling.

Now we replace $w_i$ by $T^{-1} w_i$. It remains to show that there exists a projective transformation that fixes each $[v_i]$ for $i\le n$ and sends $[v_{n+1}]$ to $[w_{n+1}]$.

We know $[w_{n+1}] = [\lambda_1 v_1 + \lambda_2 v_2 + \cdots + \lambda_n v_n]$, and we know that $\lambda_i \neq 0$ for all $i$ because our vectors are in general position. Thus the diagonal matrix with diagonal entries $\lambda_i$ is the desired transformation.

To show uniqueness, we can suppose we have 2 transformations taking one basis to another, then compose one with the inverse of another. The result will be a projective transformation that fixes a projective basis. So it will suffice to show that if a projective transformation fixes every point in a projective basis, then it is $[\lambda \cdot \text{Id}] = [\text{Id}]$.

So suppose that $[T e_i] = [e_i]$ for all $i\le n$ and that $[T(e_1 + \cdots + e_n)] = [e_1 + \cdots + e_n]$. So $Te_i = \lambda_i e_i$ for all $i\le n$. Then since $[\lambda_1 e_1 + \lambda_2 e_2 + \cdots + \lambda_n e_n] = [e_1 + \cdots + e_n]$ there is some scalar $\lambda$ with $\lambda_i = \lambda$ for all $i$. So $T = \lambda \cdot \text{Id}$ as desired.

One way to understand $\mathbb{R} P^n$ is as the union of an affine patch $\mathbb{R}^n$ with a projective hyperplane $\mathbb{R} P^{n-1}$ at infinity. This conception seems to suggest that there are two distinct kinds of points in projective space, but there is no actual difference.

Definition: An affine patch $\mathbb{A}^n$ in $\mathbb{R} P^n$ is the complement of a hyperplane $H$.

Since you can find a projective transformation sending any hyperplane to any other hyperplane, you can also send any affine patch to any other affine patch.

Definition: Let $V$ be a vector space over $\mathbb{F}$. A map $\alpha:V\to V$ is an affine map if there exist a linear map $\beta:V\to V$ and a vector $v_0 \in V$ such that $\alpha(x) = v_0 + \beta(x)$ for all $x\in V$.

In other words, an affine map is a linear map plus a constant vector.

Definition: An affine geometry is a pair $(V, \text{Aff}(V))$ where $V$ is a vector space and $\text{Aff}(V)$ is the group of affine isomorphisms of $V$.

Proposition: $PGL(\mathbb{R} P^n \setminus \mathbb{R} P^{n-1}) \cong \text{Aff}(\mathbb{A}^n)$.

Before proving the proposition we give some useful definitions.

Definition: Let $V$ be a real vector space. Then the positive projective space over $V$ is $\mathbb{S}(V) = (V\setminus 0) /( v \cong \lambda v): \lambda > 0$.

So, $\mathbb{S}(\mathbb{R}^{n+1}) = S^n$ and $\mathbb{P}(V)$ is equal to $\mathbb{S}(V) /( [v] \cong [-v])$.

Definition: $PGL^+ (V) = GL(V)/ \lambda \cdot \text{Id} : \lambda > 0$.

Now we prove the proposition in the case of 2 projective dimensions. When we make $\mathbb{R}P^2$ out of equivalence classes of Euclidean 3-space, we can represent any equivalence classes not in the $xy$ plane by points in the plane $z=1$ by writing $[x_0:x_1:x_2] = [x_0/x_2:x_1/x_2: 1]$. The points not accounted for here are of the form $[x_0:x_1:0]$ which may be considered a copy of $\mathbb{R} P^1$. So the points in the $z=1$ plane are a copy of $\mathbb{A}^2$.

Now if we consider a transformation in $PGL(\mathbb{R} P^3 - \mathbb{R} P^{2})$, we need a transformation that preserves $\langle e_1,e_2 \rangle$. So our transformation $[T]$ may be written

Where scaling allows us to set the bottom corner entry to 1. Then $T$ applied to an arbitrary $[x_0:x_1:1]$ will be a vector with 1 in the first position and the second two being an arbitrary affine transformation of $(x_0,x_1)$.

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