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.

丢番图方程是一种多项式方程，人们在其中寻求整数或有理数的解。现代研究数学家通过簇上有理点的幌子来研究此类问题，以强调问题的几何性质。要检查簇是否有有理点，首先检查是否存在实点和p进数所有素数的点 p.如果这个标准足够，就可以说哈斯原理成立。一般来说，哈斯原理可能会失败，该项目的目的是使用布劳尔-马宁障碍来研究此类失败。我们当前的目标是通过使用类似的方法对一系列表面的布劳尔-马宁障碍的存在进行分类正如 Andrew Kresch 和 Yuri Tschinkel 的论文“On the Arithemetic of del Pezzo Surfaces of Degree 2”中所述。这种分类的难度来自于我们已经使用 Tim Santens 的论文“Diagonal Quartic Surfaces with a Brauer-Manin Obstruction”中类似的方法证明的结果。这个结果意味着（宽松地），对于这些足够大的表面的任何子族，我们无法找到专门用于负责该子族中所有表面的布劳尔-马宁障碍的物体的通用公式。因此，任何此类分类都不能在整个表面系列上统一进行。