Birational geometry in positive characteristic

代数几何是研究代数簇，即一组多项式方程的解。即使代数几何开始于两千多年前，随着对圆锥曲线（如圆）几何的研究，我们仍然不能完全回答许多简单而基本的问题。例如，给定一组多项式，有许多解，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，这些情况在许多不同的领域都有有趣的应用，例如工程和生物学。由菲尔兹奖获得者S。森在20世纪80年代，是概括的主要成果分类的代数曲面，由于卡斯泰尔诺沃，恩里克和塞韦里，以高维投影品种。特别是，它预测存在一个双理性模型的任何复杂的射影品种，这是尽可能简单。在过去十年中，对森的方案开展了非同寻常的活动，因此，非常有理由期待在不久的将来取得更多的进展。这项建议的目的是研究新的方法，结合工具，在交换代数和双有理几何，这将改善这些结果，并解决一些主要的开放问题的最小模型计划在任何代数闭域。

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.