AG - Look at the curve y^2=x^3. Is it singular, integral, irreducible, reduced ? Blow-it up at the 
origin. What is the normalization of the curve - construct it. What is the genus of the smooth 
projective completion.
AT - Consider branched covers of a 2-sphere with a finite number of branching
points. What is the smallest number of branching points you need to get
something other than the two sphere. Can you give a minimal example?
AG - Consider the affine curve y^3+x^3=1. Find the genus of the smooth projective completion. 
AG - Many Riemann-Hurwitz questions were asked - I had to apply it to every other curve I wrote 
down.
AT - What is Pi_1 of a sphere minus n points or a torus minus n points. 
AG - Define elliptic curves. What can you say about them ? What are torsion points.
AG - Define an invertible sheaf. Blow up A^2 at the origin. Look at the ideal sheaf at the origin, 
is this an invertible sheaf. (Once blown-up it is)