I have very vague souvenirs about the questions. Here is what I remember :

AG - Given, x^6+1=y^3. Resolve singularities. What is the genus geom./arith. ? what kind of curve is it ? give graph
of exceptionnal curves. (incidences)
AG - What are hyperelliptic curves. Where are the ramification points.
AG - Compute h^0(P^r,O_{P^r}(k)).
AG - Take a fermat cubic, what can you say about it ? (cohomology,...)
AG - What is a del Pezzo ? examples. Properties. (same for K3) 
AT - What are the possible maps between surfaces of genus g and g'.
AT - Are there non vanishing maps between S^3 and S^3.
AT - I was also asked something about the index theorem and the relation to complex surfaces but I can't remember (I believe I also couldn't remember on that day)

Here are some practice questions they gave me earlier : 

AG - Take the Veronese embedding of P^1 inside P^n. Take the cone above it, its resolution. What is it ? (k-th
Hirzebruch surface)
AG - You have a deg 4 line bundle L over an elliptic curve. Give equation of embedding i_|L|.
AG - What is the fundamental group of a smooth quintic in P^4.
AG - Take X - -> P^N rational map. Blow it up and resolve the map. Describe morphism.
AG - What are all canonical divisors on smooth hyperelliptic curve?
AG - X, smooth hyperelliptic curve of genus 2. Give an explicit divisor of degree 0 corresponding to an element of
degree 2 torsion in the Div Class group.
AG - Take X --> C morphism, generic fiber P^1 (i.e. ruled)
	a) true or false : fibres are connected. (prove it)
	b) show it has a section
AG - Same but generic fibre is of genus 1, smooth.
AG - Classify all geometrically ruled surfaces which ar hypersurfaces in P^3.

These questions come from Kursat, Sukhendu, Rene, etc. :

AT - Define degree of map.
AG - Give double cover of P^2 ramified above a conic.
AG - Take a pencil of conics, generic enough. What is the number of singular fibres?
AG - What are the automorphism of an elliptic curve?
AG - Describe space of singular cubics. (dimension?)
AG - Compute Cl(P^1xP^1)