Rational points on varieties

丢芬图方程是一个多项式方程，人们在整数或有理数中寻求解。现代研究数学家通过在变量上的有理点来研究这类问题，以强调问题的几何性质。为了检验一个变量是否有有理点，我们首先要检验对于所有素数p是否有实点和p进点。如果这个判据是充分的，我们就说Hasse原理成立。一般来说，Hasse原理可能会失败，本项目的目的是使用Brauer-Manin障碍来研究这种失败。我们目前的目标是使用与Andrew Kresch和Yuri Tschinkel在论文“On the aritheical of del Pezzo surfaces of Degree 2”中给出的方法类似的方法对一类曲面的Brauer-Manin障碍物的存在性进行分类。这种分类的困难来自于我们已经用Tim Santens的论文“具有Brauer-Manin障碍的对角四次曲面”中类似的方法证明的结果。这一结果（宽泛地说）意味着，对于这些表面的任何一个足够大的子族，我们都不能找到一个通用公式，该公式专门用于该子族中所有表面上造成布劳尔-马宁障碍的物体。因此，任何这样的分类都不能在整个曲面族上均匀地完成。

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.