Algebraic geometry in positive characteristic: sample topics

-- Mori's bend and break argument, e.g., in the proof that every Fano variety
is covered by rational curves.

-- Deligne-Illusie's proof of degeneration of the Hodge-de Rham spectral sequence
in characteristic zero, by working modulo p^2.

-- de Jong's alterations theorem, for getting around resolution of singularities.

-- Resolution of surfaces singularities (Lipman).

-- Abhyankar's trick ("Belyi's theorem in positive characteristic") and its 
higher-dimensional analogues (see my paper "More etale covers...") Related topic:
nonabelian Artin-Schreier torsors (Nori).

-- Poonen's Bertini theorems over finite fields.

-- Automorphisms of curves in positive characteristic: examples that beat the Hurwitz
bound, bounds in positive characteristic.

-- Vector bundles over curve in positive characteristic (see Brian Osserman's thesis)

-- Nonliftable varieties, e.g., Calabi-Yau threefolds in characteristic 2 and 3 which do 
not lift to characteristic 0.

-- Mustata's ongoing work on singularity measurements in positive characteristic

-- Artin,Shioda's papers on Supersingular K3 surfaces

-- Shafarevich-Tate example of family of elliptic curves of unbounded rank 
   over F_p(t), using supersingular elliptic curves (and more recent 
   non-isotrivial examples - Ulmer?)

-- Proof of Tate's isogeny conj for char p (Zarhin?). Related: de Jong's paper on
   Barsotti-Tate groups (Tate's extension theorem in char p).