AG - Consider the equation y^3=Pi_{j=-5}^{j=5}(x-j)^(j^2) as defining a curve in A^2. 
Describe the singularities (how many, what kind are they, etc.). Does this depend on the characteristic of the base field? How? 
Now close this curve up in P^2 and describe any extra points that you get.
Are they singular? Is this curve irreducible? Why? 
Is there a nonsingular complete curve birational to this one? Is it unique? 
Find the genus of this nonsingular complete curve. 
Does every surface have a unique smooth, projective birational equivalent ? 
Give an example illustrating your assertion.
AT - Can a genus 7 curve cover a genus 2 curve? (As an unbranched cover, i.e. as a covering space)
AT - What spaces have the closed unit disk as universal cover ?
AT - Let X be a simply connected, closed, oriented 3-manifold. What can you say about its homotopy type?
AT - Let S be a Riemann surface of genus g>=2, and consider pi_1(S). Show that every \gamma<>1 in pi_1(S) has an infinite conjugacy class. [Note: this was given to me as a practice problem ahead of time, and I only had to indicate the idea of the proof in the oral exam]
AG - Show that P^2 and P^1xP^1 are not isomorphic in two different ways, one of which involves fibrations over P^1.
AG - Prove the assertion that you made in the previous problem that there are no nonconstant maps from P^2 to P^1.
AG - Suppose that X is abstractly isomorphic to P^2. How do you characterize the lines in X? How many isomorphisms X-=->P^2 are there?
AG - Is PGL(n,C) a projective variety?
AG - Over the complex numbers, what is the topological characterization of proper varieties? How generally does this hold? 
(In other words, what is it about C that makes this true?) Is the same characterization hold over Q_p?
AG - Let C_1 and C_2 be genus 12 curves and let C_1->C_2 be a morphism. What can you say? How many such morphisms are there? 
Does anything change if we consider curves of a different genus?
AG - Given two elliptic curves E_1, E_2; is there always a finite morphism E_1->E_2? How many curves are isogenous to E_1?