§ 9. Affine spaces and projective spaces

1. Definition of affine spaces

In what follows we shall confine our attention to the case where T is a left vector space over K. The dimension (over K) of the translation vector space T of an affine space E is called the dimension of E (over K) and is denoted by dim E or $\dim_K E$. An affine space of dimension one (resp. two) is called an affine line (resp. an affine plane). The elements of an affine space are also called points.

Under the conditions of Definition 1, for $t \in T$ and $a \in E$ we shall denote by $t + a$ or $a + t$ the transform of the point a under t. Then the relations

hold for $s \in T, t \in T, a \in E$. The mapping $x \mapsto x + t$ is a bijection of E onto itself, which we identify with t. Definition 1 moreover implies that, for all $a \in E$, the mapping $t \mapsto t + a$ is a bijection of T onto E. In other words, given two points $a, b$ of E, there exists one and only one translation t such that $b = t + a$; we shall denote this translation by $b - a$; then the formulae

hold for $a \in E, b \in E, c \in E$. If four points $a, b, a', b'$ of E are such that $b - a = b' - a'$, the formula

and the commutativity of addition in T show that $b' - b = a' - a$.

Given a point $a \in E$, the mapping $x \mapsto x - a$ is a bijection of $E$ onto $T$; when $E$ is identified with $T$ under this mapping, we say that $E$ is considered as the vector space obtained by taking $a$ as origin in $E$. Conversely, every vector space $T$ has canonically the structure of an affine space attached to $T$, namely the homogeneous space structure corresponding to the subgroup $\{0\}$ of $T$ (I, § 5, no. 6).

2. Barycentric calculus

3. Affine linear varieties

It amounts to the same to say that the condition of Definition 2 holds for every finite family of points of $V$.

The empty set is a linear variety; every intersection of linear varieties is a linear variety.

Let $V$ be a non-empty subset of $E$ and $a$ a point of $V$; the relation

4. Affine linear mappings

(3)

5. Definition of projective spaces

When $V = K_s^{n+1}$, we also write $\mathbf{P}_n(K)$ instead of $\mathbf{P}(K_s^{n+1})$ and $\Delta_n(K)$ instead of $\Delta(V)$.

Definition 4 can also be expressed by saying that $\mathbf{P}(V)$ is the set of lines (passing through 0) in $V$ with the origin removed; $\mathbf{P}(V)$ is therefore canonically identified with the set of lines (passing through 0) in $V$. The elements of a projective space are called the points of that space.

When $V$ is of dimension $n$, the integer $n - 1$ is called the dimension of the projective space $\mathbf{P}(V)$ if $n$ is finite, and the cardinal $n$ otherwise; this cardinal is denoted by $\dim_K \mathbf{P}(V)$ or $\dim \mathbf{P}(V)$. Thus a projective space of dimension $-1$ is empty and a projective space of dimension 0 is a single point. A projective space of dimension 1 (resp. 2) is called a projective line (resp. projective plane).

Henceforth we shall only consider left projective spaces.

6. Homogeneous coordinates

Let $V$ be a vector space of finite dimension $n + 1$ over $K$, $\mathbf{P}(V)$ the projective space of dimension $n$ derived from $V$ and $(e_i)_{0 \leq i \leq n}$ a basis of $V$. Let $\pi$ denote the canonical mapping of $V - \{0\}$ onto the quotient set $\mathbf{P}(V)$. For every point $x = \sum_{i=0}^{n} \xi_i e_i$ of $V - \{0\}$, $(\xi_0, \xi_1, \ldots, \xi_n)$ is called a system of homogeneous coordinates of the point $\pi(x)$ with respect to the basis $(e_i)$ of $V$. Every system $(\xi_i)$ of $n + 1$ elements not all zero of $K$ is therefore a system of homogeneous coordinates of a point of $P(V)$ with respect to $(e_i)$; for two such systems $(\xi_i)$, $(\xi'_i)$ to be systems of homogeneous coordinates of the same point of $P(V)$ with respect to the same basis $(e_i)$, it is necessary and sufficient that there exists an element $\lambda \neq 0$ of $K$ such that $\xi'_i = \lambda \xi_i$ for $0 \leq i \leq n$.

This definition is immediately generalized to the case where $V$ is infinite-dimensional.

Given another basis $(\tilde{e}_i)$ of $V$ such that $e_i = \sum_{j=0}^{n} \alpha_{ij} \tilde{e}_j$ ($0 \leq i \leq n$) and a system $(\xi_i)$ of homogeneous coordinates of $\pi(x)$ with respect to the basis $(e_i)$, for a system $(\bar{\xi}_i)$ of $n + 1$ elements of $K$ to be a system of homogeneous coordinates of $\pi(x)$ with respect to the basis $(\tilde{e}_i)$, it is necessary and sufficient that there exist $\lambda \neq 0$ in $K$ such that

In particular, if $e_i = \gamma_i \tilde{e}_i$ with $\gamma_i \neq 0$ ($0 \leq i \leq n$), then $\bar{\xi}_i = \mu \xi_i \gamma_i$ where $\mu \neq 0$.

7. Projective linear varieties

Let $W$ be a vector subspace of a vector space $V$; the canonical image of $W - \{0\}$ in the projective space $P(V)$ derived from $V$ is called a projective linear variety (or simply a linear variety when no confusion is to be feared); as the equivalence relation $\Delta(W)$ on $W - \{0\}$ is induced by the relation $\Delta(V)$, the projective linear variety the image of $W - \{0\}$ in $P(V)$ can be identified with the projective space $P(W)$ derived from $W$ and hence we may speak of the dimension of such a variety. In a projective space $P(V)$, the canonical image of a hyperplane (with the origin removed) of $V$ is a linear variety called a projective hyperplane (or simply a hyperplane); if $P(V)$ is of finite dimension $n$ the hyperplanes in $P(V)$ are the linear varieties of dimension $n - 1$.

Every proposition concerning vector subspaces of a vector space goes over to a proposition concerning projective linear varieties. For example, if a projective space $P(V)$ is of finite dimension $n$ and $(e_i)_{0 \leq i \leq n}$ is a basis of $V$, every linear variety $L \subset P(V)$ of dimension $r$ can be defined by a system of $n - r$ homogeneous linear equations

between the homogeneous coordinates $\xi_i$ ($0 \leq i \leq n$) of a point of $\mathbf{P}(V)$ with respect to the basis $(e_i)$, the left hand sides of (4) being linearly independent forms on V. In particular, a projective hyperplane is defined by a single homogeneous linear equation with coefficients not all zero. Conversely, the points of $\mathbf{P}(V)$ satisfying an arbitrary system of homogeneous linear equations with respect to the $\xi_i$ form a linear variety L; if the system considered consists of $k \leq n + 1$ equations, L is of dimension $\geq n - k$.

Every intersection of linear varieties of $\mathbf{P}(V)$ is a linear variety; for every subset A of $\mathbf{P}(V)$, there exists a smallest linear variety L containing A; it is called the linear variety generated by A and A is called a generating system of L.

If W is the vector subspace of V generated by $\pi^{-1}(A)$, then $L = \mathbf{P}(W)$.

If L and M are any two linear varieties in $\mathbf{P}(V)$ and N the variety generated by $L \cup M$, then (§ 7, no. 3, Corollary 3 to Proposition 4)

In particular, if $\mathbf{P}(V)$ is finite-dimensional and $\text{dim } L + \text{dim } M \geq \text{dim } \mathbf{P}(V)$, it follows from (5) that $L \cap M$ is non-empty.

Let $(x_i), (y_i)$ be two families of points in the vector space V with the same indexing set, such that $y_i = \lambda_i x_i$, where $\lambda_i \neq 0$ for all $i$. If the family $(x_i)$ is free, so is $(y_i)$ and conversely; then it is said that the family of points $\pi(x_i)$ of $\mathbf{P}(V)$ is projectively free (or simply free). It amounts to the same to say that for every index $\kappa$, the point $\pi(x_\kappa)$ does not belong to the linear variety generated by the $\pi(x_i)$ for $i \neq \kappa$. A family of points of $\mathbf{P}(V)$ which is not projectively free is called projectively related (or simply related).

For a family $(x_i)$ of points of $V - \{0\}$ to be such that the family $(\pi(x_i))$ is projectively free and generates $\mathbf{P}(V)$, it is necessary and sufficient that $(x_i)$ be a basis of V. If $\mathbf{P}(V)$ is of dimension $n$ the number of elements in such a family is therefore $n + 1$. Note that giving such a family $(\pi(x_i))$ in $\mathbf{P}(V)$ does not determine (even to within a left factor) the homogeneous coordinates of a given point of $\mathbf{P}(V)$ with respect to a basis $(y_i)$ of V such that $\pi(y_i) = \pi(x_i)$ for all $i$ (cf. no. 6).

8. Projective completion of an affine space

Let V be a (left) vector space over a field K and consider the vector space $K_s \times V$ over K; the projective space $\mathbf{P}(K_s \times V)$ is called the projective space canonically associated with the vector space V. If V is of dimension $n$, $\mathbf{P}(K_s \times V)$ is of the same dimension $n$. Consider in $K_s \times V$ the affine hyperplane $V_1 = \{1\} \times V$, whose direction (no. 3) is the subspace $V_0 = \{0\} \times V$; if a line (passing through 0) of $K_s \times V$ is not contained in $V_0$, it contains a point $(\alpha, x)$ with $\alpha \neq 0$ and $x \in V$, hence it contains also the point $\alpha^{-1}(\alpha, x) = (1, \alpha^{-1}x)$ of $V_1$; the converse is immediate and it is seen that there is a one-to-one correspondence between the points of $V_1$ and the lines (passing through 0) of $K_s \times V$ not contained in $V_0$, each of the latter meeting $V_1$ in one and only one point. It follows that the mapping $x \mapsto \phi(x) = \pi(1, x)$ is an injection (called canonical) of $V$ into the projective space $P(K_s \times V)$; $V$ is often identified with its image under this injection. The complement of $\phi(V)$ in $P(K_s \times V)$ is the projective hyperplane $P(V_0)$ called the hyperplane at infinity of $P(K_s \times V)$ (or of $V$, by an abuse of language); its points are also called the "points at infinity" of $P(K_s \times V)$ (or of $V$). If $(a_i)$ is a basis of $V$ and in $K_s \times V$ the basis is taken consisting of the elements $e_i = (0, a_i)$ and the element $e_\omega = (1, 0)$, the points at infinity in $P(K_s \times V)$ are those whose homogeneous coordinate of index $\omega$ is 0.

Let $M$ be an affine linear variety in $V$ (no. 3) and $D$ its direction; the canonical image $\phi(M)$ of $M$ in $P(K_s \times V)$ is contained in the canonical image $\overline{M} = \pi(M_2)$ of the vector subspace $M_2$ of $K_s \times V$ generated by the affine variety $M_1 = \{1\} \times M$ of $K_s \times V$. More precisely, if $(a_i)$ is an affinely free system of $M$ generating $M$, the elements $(1, a_i)$ form a basis of $M_2$ and therefore $\overline{M}$ is just the projective linear variety generated by $\phi(M)$; if $M$ is finite-dimensional, $\overline{M}$ has the same dimension as $M$. The complement of $\phi(M)$ in $\overline{M}$ is the intersection of $\overline{M}$ and the hyperplane at infinity and is equal to the canonical image $\pi(M_0)$, where $M_0 = \{0\} \times D$.

Conversely, let $N$ be a projective linear variety not contained in the hyperplane at infinity and let $R = \pi^{-1}(N)$; $R \cap V_1$ is an affine linear variety of $K \times V$ of the form $\{1\} \times M$, where $M$ is an affine linear variety of $V$ and it is immediately seen that $N$ is the affine linear variety $\overline{M}$ generated by $\phi(M)$.

There is therefore a one-to-one correspondence between the affine linear varieties of $V$ and the projective linear varieties of $P(K_s \times V)$ not contained in the hyperplane at infinity; for two affine linear varieties of $V$ to be parallel, it is necessary and sufficient that the projective linear varieties which they generate have the same intersection with the hyperplane at infinity (which is sometimes expressed by saying that the two affine linear varieties in question have the same points at infinity).

9. Extension of rational functions

If the results of no. 8 are applied to the vector space $V = K_s$ of dimension 1, it is seen that there exists a canonical injection $\phi$ of $K_s$ into the projective line $P_1(K) = P(K_s \times K_s)$; for all $\xi \in K$, $\phi(\xi)$ is the point with homogeneous coordinates $(1, \xi)$ with respect to the canonical basis (§ 1, no. 11) of $K_s \times K_s$. The complement of $\phi(K)$ in $P_1(K)$ consists of the single point with homogeneous coordinates $(0, 1)$ with respect to the above basis; it is called the "point at infinity". $P_1(K)$ is also called the projective field associated with $K$ and denoted by $\tilde{K}$, the point at infinity in $\tilde{K}$ being denoted by $\infty$.

*Consider in particular the case where $K$ is a commutative field and let f \in K(X) be a rational function in one indeterminate over K (IV, § 4); if $f \neq 0$, there is a unique expression $f = \alpha p/q$, where $\alpha \in K^*$ and $p$ and $q$ are two relatively prime monic polynomials (VII, § 1); let $m$ and $n$ be their respective degrees and let $r = \sup(m, n)$. We write

For example, if $f = 1/X$, then $\tilde{f}(0) = \infty$ and $\tilde{f}(\infty) = 0$; if

10. Projective linear mappings

Let $V, V'$ be two left vector spaces over a field $K$, $f$ a linear mapping of $V$ into $V'$ and $N = \overline{f}(0)$ its kernel. It is immediate that the image under $f$ of a line (passing through 0) in $V$ not contained in $N$ is a line (passing through 0) in $V'$; hence, on passing to the quotients, $f$ defines a mapping $g$ of $P(V) - P(N)$ into $P(V')$. Such a mapping is called a projective linear mapping (or, simply, a projective mapping); although it is defined on $P(V) - P(N)$ and not on $P(V)$ (when $N \neq \{0\}$), we shall say by an abuse of language that $g$ is a projective mapping of $P(V)$ into $P(V')$. The projective linear variety $P(N)$, where $g$ is not defined, is called the centre of $g$.

Note that, when $g$ is defined on the whole of $P(V)$ (that is when $N = \{0\}$), $g$ is an injection of $P(V)$ into $P(V')$.

When bases $(a_\lambda)_{\lambda \in L}, (b_\mu)_{\mu \in M}$ are given in $V$ and $V'$ respectively, a projective mapping of $P(V)$ into $P(V')$ maps a point of $P(V)$ with homogeneous coordinates $\xi_\lambda$ ($\lambda \in L$) to a point of $\mathbf{P}(V')$ with a system of homogeneous coordinates $\eta_\mu$ ($\mu \in M$) of the form

The centre of $g$ is the linear variety defined by the equations

If $C$ is the centre of $g$ and $M$ is a linear variety of $\mathbf{P}(V)$, the image under $g$ of $M - (M \cap C)$ is a linear variety of $\mathbf{P}(V')$ denoted (by an abuse of language) by $g(M)$. Then

(§ 7, no. 4, formula (12)). If $M'$ is a linear variety of $\mathbf{P}(V')$, $g^{-1}(M') \cup C$ is a linear variety of $\mathbf{P}(V)$ and

It is said, by an abuse of language, that $g^{-1}(M') \cup C$ is the inverse image of $M'$ under $g$.

As the values taken by a linear mapping on a basis $(e_i)$ of $V$ can be chosen arbitrarily in $V'$, it is seen that there exists a projective linear mapping of $\mathbf{P}(V)$ into $\mathbf{P}(V')$ taking arbitrary values at the points $\pi(e_i)$. But (even when $g$ is everywhere defined) giving $g(\pi(e_i))$ does not determine $g$ uniquely (Exercise 10).