Math 480, Spring 2014, Assignment 11

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

1. Dehomogenization (of an ideal $I\subseteq\mathsf{k}[x_0,\dots,x_n]$ at $x_0$).
2. Homogenization (of an ideal $I\subseteq\mathsf{k}[x_1,\dots,x_n]$ with respect to $x_0$).
3. Graded monomial order.
4. Homogenization (of a monomial order on $\mathsf{k}[x_1,\dots,x_n]$).

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

1. Theorem describing generators for the homogenization of an ideal (Theorem 4 in section 8.4).
2. Theorem describing generators for the dehomogenization of an ideal (this rather easy result is not in the text).
3. Theorem relating the variety of a homogenization to the Zariski closure of (the image of) an affine variety in projective space. (Confusingly, the book takes the theorem as a definition, then treats our definition as a theorem -- see Definition 6 and Proposition 7 in section 8.4.)
4. Theorem relating the variety of a dehomogenization to an "affine patch" of a projective variety (this easy result is not in the text).

Solve the following problems:

1. Section 8.4, problems 2, 3, and 6.

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