Homework Assignment 1

Let $K$ be a field contained in $\mathbb R$, and fix a number $r > 0$ in $K$. Recall that $\mathbb P^2(K)$ is the projective plane, while $S^2(K)$ is the sphere of radius $r$:
$S^2(K) = \left \{ (u,v,w) \in \mathbb A^3(K) \ \biggl| \ u^2 + v^2 + w^2 = r^2 \right \}.$
This assignment is meant to show $\mathbb P^2(K)$ is not the same as $S^2(K)$. This assignment is due Friday, September 6, 2013 at the start of class.

Problem 1. For each point $(\tau_1 : \tau_2 : \tau_0) \in \mathbb P^2(K)$ satisfying $\tau_0 \neq 0$, consider the line
L: \qquad \begin{aligned} \tau_0 \, u + \tau_1 \, w & = \tau_1 \, r \\ \tau_0 \, v + \tau_2 \, w & = \tau_2 \, r \end{aligned}
Define a map $\phi: \mathbb P^2(K) \to S^2(K)$ by considering where the line $L$ intersects the sphere $S^2(K)$:
$\phi: \qquad (\tau_1 : \tau_2 : \tau_0) \qquad \mapsto \qquad \left( \dfrac {2 \, \tau_1 \, \tau_0}{\tau_1^2 + \tau_2^2 + \tau_0^2} \, r, \ \dfrac {2 \, \tau_2 \, \tau_0}{\tau_1^2 + \tau_2^2 + \tau_0^2} \, r, \ \dfrac {\tau_1^2 + \tau_2^2 - \tau_0^2}{\tau_1^2 + \tau_2^2 + \tau_0^2} \, r \right).$
Show that $\phi$ is a well-defined map.

Problem 2. Show that $\phi$ is surjective. Hint: Show $\phi(P) = (u,v,w) \neq (0, 0, r)$ if and only if $P = ( u : v : r-w)$.

Problem 3. Verify the following set equalities and isomorphisms:
\begin{aligned} U_1 & := \left \{ P \in \mathbb P^2(K) \ \biggl| \ \phi(P) \neq (0, \, 0, \, r) \right \} = \left \{ (\tau_1 : \tau_2 : \tau_0) \in \mathbb P^2(K) \ \biggl| \ \tau_0 \neq 0 \right \} \simeq \mathbb A^2(K), \\[5pt] U_2 & := \left \{ P \in \mathbb P^2(K) \ \biggl| \ \phi(P) = (0, \, 0, \, r) \right \} = \left \{ (\tau_1 : \tau_2 : \tau_0) \in \mathbb P^2(K) \ \biggl| \ \tau_0 = 0 \right \} \simeq \mathbb P^1(K). \\[5pt] \end{aligned}
Conclude that $\mathbb P^2(K) \simeq \mathbb A^2(K) \cup \mathbb P^1(K)$.

Problem 4. Show that $\phi$ is injective on $U_1$, i.e., if $\phi \bigl( (\tau_1 : \tau_2 : \tau_0) \bigr) = \phi \bigl( (\omega_1 : \omega_2 : \omega_0) \bigr)$ then $(\tau_1 : \tau_2 : \tau_0) = (\omega_1 : \omega_2 : \omega_0)$.

Problem 5. Show that $\phi$ is not injective on $U_2$. Conclude that $\mathbb P^2(K)$ is not the same as $S^2(K)$.