Math 480, Spring 2013, Assignment 10

Note: this material will not appear on the final exam.

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

1. Point derivation (of the polynomial ring \(k[x_1,\dots,x_n]\) at a point \(p=(p_1,\dots,p_n)\in k^n\)).
2. Tangent space (to \(k^n\) at a point \(p\)).
3. Tangent space (to a variety \(V=\mathbb{V}(I)\) at a point \(p\in V\)).
4. Singular point (of a variety \(V\)--you may leave the symbol \(\text{dim}(V)\) undefined until you have ready Chapter 9).

Do the following problems:

1. Consider the variety \(V=\mathbb{V}(x^2-y^2)\in k^2\). Draw a picture of this variety, then see if you can guess any singular points. Confirm your guess by calculating the dimension of the tangent space at each point.
2. Consider the variety \(V=\mathbb{V}(y^2-x(x-1)^2)\). Draw a picture of this variety (reviewing the calculus of implicit functions may be helpful here!). Then proceed as above.
3. Consider the variety \(V=\mathbb{V}(y^2+2y+1-x^3)\). Proceed as above.
4. Consider a plane curve of the form \(C=\mathbb{V}(p)\) where \(p\) is any polynomial of two variables. Try to find some criterion which identifies the singular points of \(C\).
5. Try to generalize the criterion you found above to varieties of arbitrary dimension.
6. On the first page of the course syllabus there are pictures of three surfaces in \(k^3\), along with their defining equations. Try to find all singular points of these surfaces.