Birational geometry in positive characteristic

代数几何是对代数变量的研究，即多项式方程集合的解集。尽管代数几何在两千多年前就开始了，随着对二次曲线（如圆）几何的研究，我们仍然不能完全回答许多简单和基本的问题。例如，给定一个多项式集合，有有限个解，我们想知道这些解的个数。我们在许多特殊情况下都知道这个数字，这些情况在许多不同的领域都有有趣的应用，比如工程和生物学。最小模型计划的目标是将Castelnuovo、Enrique和Severi等人提出的代数曲面分类的主要结果推广到高维投影曲面。该计划由菲尔兹奖得主S. Mori在20世纪80年代发起。特别是，它预测了任何复杂射影变化的两国模型的存在，这是尽可能简单的。过去10年里，人们对森喜朗的计划进行了非同寻常的研究，我们完全有理由预计，在不久的将来会取得更大的进展。本建议的目的是研究新的方法，结合交换代数和二分几何的工具，这将改善这些结果和解决一些主要的开放问题的最小模型规划在任何代数封闭的领域。

Algebraic geometry is the study of algebraic varieties, i.e. the sets of solutions of a collection of polynomial equations. Even though algebraic geometry started more than two thousand years ago, with the study of the geometry of conic sections, such as circles, we still cannot completely answer many simple and fundamental questions. For example, given a collection of polynomials, with finitely many solutions, we would like to know the number of these solutions. We know this number in many special cases, and these cases have interesting applications in many different fields, such as engineering and biology. The goal of the Minimal Model Programme, started by Fields medalist S. Mori in the 1980s, is to generalise the main results of the classification of algebraic surfaces, due to Castelnuovo, Enrique and Severi, to higher dimensional projective varieties. In particular, it predicts the existence of a birational model for any complex projective variety which is as simple as possible. The last decade has seen exceptional activity towards Mori's programme, and it is very reasonable to expect that much more progress will be made in the near future. The aim of this proposal is to study new methods, by combining tools in commutative algebra and birational geometry, which would improve these results and solve some of the main open problems in the Minimal Model Programme over any algebraically closed field.