Birational Algebraic Geometry in Characteristic Zero and Positive Characteristic

代数几何是数学中最古老和最活跃的领域之一。它在交换代数、数论、微分几何、弦论等许多邻近领域中起着重要的作用。代数几何研究多项式方程的解集。这些解集定义了几何对象，重要的是对它们进行分类并理解它们的定性行为。双有理几何旨在对这些几何对象进行分类（直到双有理同构）。这是一个主要领域的代数几何，近年来一直在中心的壮观的进展。许多重要的核心问题仍然存在。PI希望利用这些最新的成功，以取得进展，对其余的关键开放的问题和programmes.首席研究员将进行研究代数几何，特别是在高维双有理代数几何的特征0和特征p 0代数闭域。特别是这个项目的重点是有关的问题的最小模型程序的threefolds在一个代数闭域的特征p = 2，3，5包括有关问题的逆的附加，和最小模型程序和奇点的threefolds在一个代数闭域的特征p 0。PI还计划研究一般消失定理和定义在特征p的代数闭域上的变种的双有理几何，以及与广义对数正则对的双有理几何和维度大于或等于4的对数翻转的终止有关的问题。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic Geometry is one of the oldest and most active areas of mathematics. It plays an important role in many neighboring fields such as Commutative Algebra, Number Theory, Differential Geometry, and String Theory to name a few. Algebraic Geometry studies the solution sets of polynomial equations. These solution sets define geometric objects and it is important to classify them and understand their qualitative behavior. Birational Geometry aims to classify these geometric objects (up to birational isomorphism). This is one of the principal areas of Algebraic Geometry which in recent years has been at the center of spectacular progress. Many important central questions remain. The PI hopes to capitalize on these recent successes to make progress towards the remaining key open problems and conjectures.The principal investigator will conduct research in Algebraic Geometry and especially in Higher Dimensional Birational Algebraic Geometry over algebraically closed fields of characteristic 0 and characteristic p 0. In particular this project is focused on questions related to the minimal model program for threefolds over an algebraically closed field of characteristic p = 2,3,5 including related questions on the inversion of adjunction, and to the minimal model program and the singularities of threefolds over an algebraically closed field of characteristic p 0. The PI also plans to investigate generic vanishing theorems and the birational geometry of varieties defined over an algebraically closed field of characteristic p 0 as well as questions related to the birational geometry of generalized log canonical pairs and to the termination of log-flips in dimension greater or equal than 4 (in characteristic 0).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.