AG:

1) Show that P^1 ^(n)(n-th symmetric product) is isomorphic to P^n.
2) Take S=P^1 x C, g(C)=1 (C-smooth) (can try to do with g=2, too)
Take a (rk 2) vector bundle,V, on S, which sits in a ses
0 -> K -> V -> O_S ->0
(on S) How many of these are there? Take X=P(V) ->C, the fibres over a
pt,x in C are surfaces - which ones? Do they depend on the point,x?
Do it for each extension,V.
(You get Hirzebruch surfaces - which ones.Bundles on P^1, extensions,etc.
come on the way)
3) You cannot have a surj.map P^n -> P^k, n>k. Can you have a map which is
not constant? Can the image be 2 pts.?

4) Intersection theory on P^1 x P^1 , which divisors there are effective,
how do you check a divisor is effective? Want to get from P^1 x P^1 to P^2 by
blowing up and down - can we blow it down? (Are there curves with
negative self intersection on P^1xP^1? Castelnuovo's contractibility
criterion). Relation between self intersection
of a curve,slf-int of   its proper transform and of its total transform
(two of them are equal - which two?). Describe the procedure of getting
from P^1xP^1 to P^2 - blow up an intersection of the ruling, and blow down
the stict transform of the two curves of the ruling. Show the image of the
exceptional divisor from the first blow-up is effective, with self-int 1;
what's the dim of the linear system corresp.to it (2), why is it base-pt
free?

5) Take a curve of genus 2,C, and a degree 2 map from a curve, D, to C.
How many such exist - g(D)=2, ramif. at two points. What determines the
map- have a rep. of pi_1 of (C-p-q) to Z/2. How many such do we have
(the map factors through the abelinization, H_1(C-p-q,Z)= H_1 (C)= Z^4,
so we have 16 of these)

Math Phys:
Supersymmetry, Super Lie Algebras, Poincare and Lorentz groups,
 multiplets in N=(1,1) supersymmetry.