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What are the most well known results in classical (à la Weil) algebraic geometry in characteristic $p$, which are thought to be true (but not yet proved) in characteristic 0?

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The first two things that pop into my head are both proved in characteristic $0$, but off the top of my head I'd guess that they both only have proofs that must be done in positive characteristic. The first is that if there is an irreducible curve $C$ on $X$ such that $C.K_X<0$, then $X$ contains a rational curve (a.k.a. Mori's "Bend and Break" argument). The other is the degeneration of the Hodge de-Rham s.s. for $X/k$ of characteristic $0$ and hence having Hodge symmetry in characteristic $0$. –  Matt Oct 16 '12 at 0:04
    
should maybe made community wiki –  Julian Kuelshammer Nov 9 '12 at 22:12

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