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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?

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Let $X$ be a non-singular hypersurface of degree $d$ in $\mathbb{P}^n$ over a number field $k$. If $d > n+1$, then $X$ has general type. The Bombieri-Lang conjectue therefore predicts that the rational points on $X$ are not Zariski dense. So in general, one should expect very few rational points and perhaps one would expect counterexamples to the Hasse principle to occur.

Now comes the embarrassing part. There is not a single known example of a hypersurface of general type with $n>2$ which fails the Hasse principle!

One difficulty in producing counter examples when $n>3$ is that the Brauer group consists of constant algebras and moreover such hypersurfaces are simply connected. So one cannot hope to produce counterexamples to the Hasse principle via the Brauer-Manin obstruction or étale descent. It is still expected that counterexamples to the Hasse principle should occur, so in particular that the Brauer-Manin obstruction is not the only one.

Check out the paper:

Poonen - The Hasse principle for complete intersections in projective space.

See also the appendix by Colliot-Thélène in the paper

Poonen, Voloch - Random diophantine equations.

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  • $\begingroup$ I thought that "general type" refers to a Zariski-open set in the moduli space of degree $d$ hypersurfaces? For instance the Fermat hypersurface $a_1 x_1^d + \cdots + a_n x_n^d = 0$ is surely not 'general' for any $d$. $\endgroup$
    – Stanley Yao Xiao
    Commented Apr 10, 2014 at 22:38
  • $\begingroup$ You're thinking of "generic", not "general type", which has a specialized technical meaning in algebraic geometry. See e.g. en.wikipedia.org/wiki/Surface_of_general_type for the case of surfaces. $\endgroup$
    – Alison Miller
    Commented Apr 11, 2014 at 0:08

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