Obstruction Theory Of Rational Points On Varieties

In 1970,Manin constructed an intermediate set between X(k)and X(Ak)via the theory of Brauer group.Now,it's called Brauer-Manin obstruction.Usually,this set is better than the set of rational points to judge whether it is non-empty.Using the theory of principle homogeneous spaces on algebraic groups,another kind of obstruction can be constructed,which is called descent obstrution.In 1999,Skorobogatov constructed a nice algebraic varieties,whose Manin obstrution set is a non-empty set but the set of rational points is empty set.In fact,in that example,the descent obstruction is empty set,so in general,obstrution is not unique.But to make matter worse,in 2010,Poonen constructed an example of algebraic varieties where descent obstruction is also non-empty set,but its rational points set is empty.This article is a summary report.We will mainly introduce the basic theory of Manin obstrution and descent obstrution,and explain in detail about the example of Skorobogatov.