Math 480, Spring 2014, Assignment 8

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

1. Orbit variety (for a given finite matrix group $G$ -- your book denotes this object by $V_F$.)
2. Projective $n$-space (over a field $\mathsf{k}$).
3. Point (of projective $n$-space).
4. Line (in projective space).
5. $d$-plane (in projective space).
6. Hyperplane at $\infty$ (in projective space).
7. Homogeneous coordinates (of a point in projective space).

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

1. Theorem on the invariance of the orbit variety (Corollary 8 in section 7.4).
2. Theorem relating orbits to points of the orbit variety (Theorem 10 in section 7.4).
3. Theorem on the intersection of two lines in $\mathbb{P}^2(\mathsf{k})$.

Solve the following problems:

1. Section 7.4, problems 7 and 13.
2. Section 8.1, problems 2, 4, and 10.

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