Rational points on varieties

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A Diophantine equation is a polynomial equation where one seeks solutions in the integers or the rational numbers. Modern research mathematicians study such problems through the guise of rational points on varieties, in order to emphasise the geometric nature of the problem.To check whether a variety has a rational point, one first checks whether there is a real point and a p-adic point for all primes p. If this criterion is sufficient one says that the Hasse principle holds. In general the Hasse principle can fail, and the aim of this project is to study such failures using the Brauer-Manin obstruction.Our current objective is to categorise the existence of the Brauer-Manin obstruction of a family of surfaces by using a similar method as given in the paper "On the Arithemetic of del Pezzo Surfaces of Degree 2" by Andrew Kresch and Yuri Tschinkel. The difficulty of this categorisation comes from a result we have already proved using a similar method as in the paper "Diagonal Quartic Surfaces with a Brauer-Manin Obstruction" by Tim Santens. This result implies (in loose terms) that for any sub-family of these surfaces that is sufficiently large we cannot find a general formula for an object that specialises to the object responsible for the Brauer-Manin obstruction for all surfaces in this sub-family. Therefore any such categorisation cannot be done uniformly over the whole family of surfaces.