The Hasse principle is perhaps an at-first naive generalization of the Chinese remainder theorem; that if a linear equation can be solved modulo $p$ for any prime $p$, then it can be solved in the integers. The first significant result is the Hasse-Minkowski theorem, which asserts that the local to global principle holds for representing zero by quadratic forms over number fields. The Hasse Principle is known to fail, however, the simplest example given by Selmer: $3x^3 + 4y^3 + 5z^3 = 0$ has a solution over $\mathbb{R}$ and in all $p$-adic fields, but no non-trivial solutions over the rationals.

Given that the Hasse principle is not expected to be true in general, the better question is to ask when can it fail. There is an obvious obstruction to the Hasse principle, known as the Brauer-Manin obstruction. It seems that there is a lot of interest in investigating whether the Brauer-Manin obstruction is the only obstruction.

By an influential paper of A.N. Skorobogatov in 1999, there are examples where the Hasse principle fails but which cannot be explained by the Brauer-Manin obstruction. In the same paper he mentioned a result due to Sarnak and Wang who gave a counter example of degree 1130 in $\mathbb{P}_\mathbb{Q}^4$, conditioned on a conjecture of Lang. Skorobogatov remarked that his approach does not account for the Sarnak/Wang example.

The degree 1130, which is large relative to the dimension of the ambient space (being 4 in this case), seems striking to me. Is the expectation that the Brauer-Manin obstruction is insufficient to account for the failure of Hasse principle for varieties whose degrees are much larger than their dimension?