Question

What is the state-of-the-art of the proof/counterexamples of the Hasse principle for high-dimensional hypersurfaces (say 3-folds or more)?

Answer 1

Regarding counterexamples, there is Sarnak-Wang, and then the results of Bjorn Poonen. Regarding proofs that the Hasse principle holds, I don't know the best result over ALL global fields. Of course global function fields are $C_2$, thus the Hasse principle trivially holds over global function fields for hypersurfaces of degree $d$ in $\mathbb{P}^n$ satisfying $d^2 \leq n$, because these hypersurfaces have rational points.

Edit. I should add that some experts have asked / conjectured about whether the Brauer-Manin obstruction is the only obstruction to the Hasse principle for every rationally connected, smooth, projective variety over a global field. Since the Brauer-Manin obstruction vanishes for smooth hypersurfaces of dimension $\geq 3$, in particular these people are asking whether the Hasse principle holds for hypersurfaces of degree $d$ in $\mathbb{P}^n$ with $d\leq n$ and $n\geq 4$. To my knowledge there are no known counterexamples.

Answer 2

A recent positive result on cubic hypersurfaces:

http://arxiv.org/abs/1207.2385