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.

Homework Assignment 1 Download

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).

Advertisements

About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
This entry was posted in MA 59800 and tagged , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s