In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'.
Saying this more technically, a thin set of type II is any subset of where V′ satisfies the same assumptions as V and φ is generically surjective from the geometer's point of view.
At the level of function fields we therefore have While a typical point v of V is φ(u) with u in V′, from v lying in V(K) we can conclude typically only that the coordinates of u come from solving a degree e equation over K. The whole object of the theory of thin sets is then to understand that the solubility in question is a rare event.
[1] A result of S. D. Cohen, based on the large sieve method, extends this result, counting points by height function and showing, in a strong sense, that a thin set contains a low proportion of them (this is discussed at length in Serre's Lectures on the Mordell-Weil theorem).
Being Hilbertian is at the other end of the scale from being algebraically closed: the complex numbers have all sets thin, for example.
[10] In fact Colliot-Thélène conjectures WWA holds for any unirational variety, which is therefore a stronger statement.
This conjecture would imply a positive answer to the inverse Galois problem.