Math 480, Spring 2013, Assignment 9

Carefully define the following terms, then give one example and one non-example of each:

1. Projection map.
2. Zariski closure.
3. Polynomial map.
4. Rational map.
5. Graph of a map.

Carefully state the following theorems (you need not prove them):

1. Geometric extension theorem.
2. Closure theorem.
3. Polynomial implicitization theorem (Theorem 3.3.1).
4. Rational implicitization theorem (Theorem 3.3.2).

Do the following problems:

1. Let \(S\) be the image of the polynomial map given by \((x, y, z) = (uv, uv^2, u^2)\). Find the ideal of the Zariski closure of \(S\). Then (working over \(\mathbb{C}\)) find all the points of the Zariski closure of \(S\) which do not lie in \(S\) itself.