Question 1 (7 points):

Let be an algebraic closed field, and let and be algebraic sets. Let be a polynomial map and let be the corresponding map between coordinate rings.

  1. Prove that if is surjective, then is injective.
  2. Prove that if is surjective, then is injective.
  3. Show that the converses of parts 1 and 2 don’t hold.
  4. If is an algebraic subset, prove that is also an algebraic subset.
  5. If is an algebraic subset, give an example to show that need not be an algebraic subset.

Hints: Two polynomials are the same if and only if they take on the same value at every point. Two points and are different if and only if there is a polynomial that takes on different values at and . For 3 and 5, consider the map given by .

Question 2 (3 points):

The nodal cubic is the algebraic set .

  1. Show that the map given by lands inside .
  2. Show that the map is surjective onto . Where does fail to be injective?
  3. Prove that is irreducible. Using or might help…

Note (not necessary to do the problem, but fun and potentially helpful)

Sketch the real part of (maybe by considering how it’s related to the graph of ).

The map has the following geometric interpretation, which may be enlightening. Consider the line through the origin with slope , i.e., . If we restrict to this line, it will be a cubic polynomial in one variable, and hence have three roots. The origin will always be a double root, and hence there will be one more root. Geometrically, this means the line intersects at the origin with multiplicity two, and at one other point. This other point is .