Math 480, Spring 2013, Assignment 2

Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.

- F. Klein

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

1. Affine variety (defined by the polynomials \(f_1,\dots,f_s\)).
2. Rational function (of \(t_1,\dots,t_m\)). (Note that it would be better to call these objects rational expressions. Why?)
3. Polynomial parametrization (of an affine variety).
4. Rational parametrization (of an affine variety).
5. Implicit representation (of an affine variety).
6. Implicitization problem.

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

1. Theorem on unions and intersections of affine varieties (Lemma 2 in section 1.2).

Solve the following problems:

1. Working in \(\mathbb{R}^2\), sketch the variety \(\mathbf{V}(x^2-y^2)\).
2. Working in \(\mathbb{R}^3\), sketch the variety \(\mathbf{V}(xz^2-xy)\).
3. Working in \(\mathbb{R}^3\), sketch the variety \(\mathbf{V}(x, z^2-y)\).
4. Working in \(\mathbb{R}^3\), sketch the variety \(\mathbf{V}(x-1, y-2, z-3)\).
5. Show that any single-point set \(\{(a_1,\dots,a_n)\}\subset k^n\) is an affine variety.
6. Show that any finite subset of \(k^n\) is an affine variety. (Hint: use the previous problem together with the theorem on unions and intersections of affine varieties.)
7. Show that \(\mathbb{Z}\subset\mathbb{R}^1\) is not an affine variety. (Hint: suppose to the contrary that \(\mathbb{Z}\) is the variety defined by the polynomials \(f_1,\dots,f_s\). Then each \(f_i\) is a polynomial that vanishes at every integer. What can you conclude about \(f_i\)?) In light of the previous problem, why doesn't this contradict the theorem on unions and intersections?
8. Find an implicit representation of the affine variety parametrized by \(x = \frac{t}{1+t}, y = 1 - \frac{1}{t^2}\). Does the parametrization cover every point of this variety?

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Questions:

• I am slightly confused over the notion of a non-example of an affine variety. Would this be the affine variety of a polynomial with no roots? - --Rob Moray

Well, non-example isn't exactly a precise term; I suppose one could argue that a chair is a non-example of an affine variety. But the spirit of the question is to give a subset of \(k^n\) which is not an affine variety, i.e. not definable by any system of polynomial equations -- see problem 7. (The empty set is not a non-example since it is the variety \(\mathbf{V}(\{1\})\).) - Steven.Jackson (talk) 21:25, 5 February 2013 (EST)

• I wanted to also confirm that we have only covered 1.1 and 1.2 in class thus far. I see that some of these problems are covered in 1.3 (Parameterizations of Affine Varieties), but I dont remember covering these in class. Should I just be focusing on the problems that are covered in the first two sections for Tuesday? - Rob Moray 11:44, 9 February 2013 (EST)

More or less. The only problem from 1.3 is the last one, which is worth trying but I don't think it'll be the quiz question. We'll talk a bit more about it on Tuesday. -Steven.Jackson (talk) 19:31, 11 February 2013 (EST)

• Is the only difference between problems 5 & 6 the number of points? In #5 the number of points, k = the dimension of the affine space and in #6 it's any n?

Actually in number 5 there is only a single point (with \(n\) coordinates), while in number six the number of points is arbitrary (and unrelated to the dimension of the affine space). But that's the only difference. -Steven.Jackson (talk) 07:27, 12 February 2013 (EST)