Rational points on varieties

品种理性点

• 批准号: 2889566
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2889566
• 关键词: Rational; points; varieties

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.

二磷剂方程是一个多项式方程，其中人们在整数或理性数字中寻求解决方案。现代研究数学家通过品种的理性观点来研究此类问题，以强调问题的几何性质。要检查一个品种是否具有理性点，首先检查所有素数是否有真实的观点和p-adic点。如果此标准足够，则说Hasse原则将持有。通常，Hasse原理可能会失败，该项目的目的是使用Brauer-Manin障碍物研究此类失败。您目前的目标是通过使用论文中给出的类似方法对表面家族的Brauer-Manin障碍物的存在进行分类。这种分类的困难来自于我们已经证明使用类似的方法，与蒂姆·桑坦斯（Tim Santens）的“对角线四分之一表面”具有类似的方法。该结果意味着（用宽松的术语），对于这些表面的任何子家庭，我们都找不到一个对象的通用公式，该物体专门针对该亚家族中所有表面的物体。因此，任何这样的分类都不能在整个表面上统一地进行。