Exercise 18, § 9

Let $E = \mathbf{P}(V)$ be a projective plane over a field $K$ and $u$ a bijective mapping of $E$ onto itself, arising when passing to the quotients from a bijective semi-linear mapping $v$ of $V$ onto itself, relative to an automorphism $\sigma$ of $K$.

(a) Show that the four following properties are equivalent: ($\alpha$) for all $x \in E$, $x, u(x)$ and $u^2(x)$ lie on a straight line; ($\beta$) every line of $E$ contains a point invariant under $u$; ($\gamma$) through every point of $E$ there passes a line invariant under $u$; ($\delta$) for every line $D$ of $E$, the three line $D, u(D)$ and $u^2(D)$ have a common point. (Show first that ($\alpha$) and ($\gamma$) are equivalent; deduce by duality (Exercise 12) that ($\beta$) and ($\delta$) are equivalent; prove finally that ($\gamma$) implies ($\beta$) and deduce by duality that ($\beta$) implies ($\gamma$).)

(b) Suppose that $u$ has the properties stated in (a). Show that if there exists in $E$ a line $D$ invariant under $u$ and containing only a single point invariant under $u$, $u$ arises from a transvection $v$ of $V$ when passing to the quotients ($§ 10$, Exercise 11). (Show that every line invariant under $u$ contains $a$; by considering a line not passing through $a$, show that there exists a line $D_0$ passing through $a$ and containing at least two points invariant under $u$; conclude that all the points of $D_0$ are necessarily invariant under $u$.)

(c) Suppose that $u$ has the properties stated in (a). Show that if there exists in $E$ a line $E$ invariant under $u$ and containing only two points invariant under $u$, $u$ arises from a dilatation $v$ of $V$ when passing to the quotients ($§ 10$, Exercise 11). (If $a, b$ are the two points of $D$ invariant under $u$, show that every line invariant under $u$ passes through $a$ or through $b$; then note that there exist at least two other points $c, d$ distinct from $a, b$ and invariant under $u$ and therefore the line $D_{cd}$ passes through $a$ or $b$; conclude by proving that all the points of $D_{cd}$ are invariant under $u$.)

(d) Suppose that $u$ has the properties stated in (a) and that every line of $E$ invariant under $u$ contains at least three distinct points invariant under $u$; then there exists in $E$ a projective basis (Exercise 10) each point of which is invariant under $u$; conclude that there exists a basis $(e_i)_{1 \leq i \leq 3}$ of $V$ such that $u$ arises from a semi-linear mapping $v$ of $V$ into itself such that $v(e_i) = e_i$ for $1 \leq i \leq 3$, when passing to the quotients. The set of points of $E$ invariant under $u$ is then the projective plane $\mathbf{P}(V_0)$, where $V_0$ is the vector space over the field $K_0$ of invariants of $\sigma$, generated by $e_1, e_2, e_3$.

(e) Suppose henceforth that $u$ satisfies the conditions of (a) and (d) and that neither $u$ nor $u^2$ is the identity. Show that there exists $\gamma \in K$ such that $\gamma^\sigma = \gamma$ and

(1)

for all $\xi \in K$ (use conditions ($\alpha$) of (a) and the existence of $\zeta \in K$ such that $\zeta^\sigma \neq \zeta$). Show that $\gamma \neq -1$ and that, for all $\xi \in K$ such that $\xi^\sigma \neq \xi$,

(2)

(apply (1) replacing $\xi$ by $\xi^2$); extend (2) to all $\xi \in K$ by noting that $\xi = \eta - \zeta$, where $\eta^\sigma \neq \eta$ and $\xi^\sigma \neq \zeta$. Conclude that $\gamma \neq 1$, then deduce from (1) and (2) that

(3)

for all $\xi \in K$ and that $\gamma + \gamma^{-1}$ belongs to the centre of $K$. Obtain the converse.

Give an example where $\gamma$ does not belong to the centre of $K$, but where $\gamma + \gamma^{-1}$ is in this centre.