New perspectives for canonical dimension

在一个代数簇上寻找有理点相当于在给定的域上寻找系数为多项式方程组的解。许多关于二次型、中心单代数或更一般的伽罗瓦上同调的问题，可以表述为线性代数群下的某个投射齐次簇是否有有理点。典范维数度量了一个变种离有一个合理点的距离，因此比仅仅考虑合理点的存在允许变种之间更精细的区分。相应地，这个概念提供了关于二次型和中心单代数的有价值的信息。到目前为止，标准维数的概念主要用于投射齐次簇的上下文中。它的计算通常涉及受代数拓扑启发的工具，特别是上同调运算和motivic分解。这个项目沿着沿着两个方向组织： (a)在线性代数群领域之外的新环境中使用规范维数和相关方法。这些技术可能是有用的，每当一个试图找到合理的点上的代数品种，这种情况当然不限于研究射影齐次品种，但实际上对应于很大一部分算术几何。(b)利用代数几何中其他部分的方法计算射影齐次簇的标准维数。这包括从经典代数几何，这是经常使用的算术几何，和Chow-Witt理论的方法，这是目前用于研究投射模的技术。明确的问题，将被调查： (1)有限p-群在特征不同于p的域上的仿射空间上的作用能固定一个有理点吗？ (2)计算Severi-Brauer簇的标准维数。 (3)获得二次型分裂模式的新信息。 (4)证明塞尔消失猜想的二次精化。

Finding rational points on an algebraic variety amounts to finding solutions of systems of polynomials equations with coefficients in a given field. Many questions concerning quadratic forms, central simple algebras, or more generally Galois cohomology, may be formulated as whether a certain projective homogeneous variety under a linear algebraic group has a rational point or not. Canonical dimension measures how far a variety is from having a rational point, and thus allows finer distinctions between varieties than the mere consideration of the existence of rational points. Accordingly, this notion provides valuable informations on quadratic forms and central simple algebras. Until now, the concept of canonical dimension has been mostly used in the context of projective homogeneous varieties. Its computation often involves tools inspired by algebraic topology, in particular cohomological operations and motivic decompositions.This project is organised along two directions: (a) Use canonical dimension and the associated methods in new contexts, outside the realm of linear algebraic groups. These techniques may be useful whenever one is trying to find rational points on an algebraic variety, a situation which is of course not limited to the study of projective homogeneous varieties, but in fact corresponds to a large part of arithmetic geometry.(b) Use methods coming from other parts of algebraic geometry to compute the canonical dimension of projective homogeneous varieties. This includes techniques from classical algebraic geometry, which are routinely used in arithmetic geometry, and Chow-Witt-theoretic methods, which are currently used for the study of projective modules.Explicit problems that will be investigated are: (1) Does the action of a finite p-group on the affine space over a field of characteristic different from p fix a rational point? (2) Compute the canonical dimension of the Severi-Brauer varieties. (3) Obtain new informations on the splitting patterns of quadratic forms. (4) Prove a quadratic refinement of Serre's vanishing conjecture.