Research in Birational Geometry

本研究方向为代数几何，涵盖与Mori理论、曲线模、几何不变量理论和环面几何等领域相关的几何基础问题。第一个主要问题是曲线的模空间是否无规的经典问题。对于Mori理论来说，这是一个非常自然的问题，它确实是为了研究不均匀性问题而发明的，但令人惊讶的是，迄今为止还没有人从这个方向进行认真的尝试——也许是因为该领域的主要工作是在Mori理论发明之前完成的。申请人认为Mori理论提供了重要的新见解，并以许多具体的例子来支持这一点。第二个建议的方向是二分几何的基础工作。申请人与胡毅一起观察到Mori理论与几何不变量理论之间的基本联系。他们将森梦空间抽象地定义为具有最好的森理论性质的空间。这样的例子有很多。他们将任何品种与一个自然环联系在一起，他们称之为Cox环，并表明Mori梦空间的特征是这个环的有限代，而且，在某种程度上推广了Cox关于环型品种作为仿射空间商的构造，以一种自然的方式是相关仿射品种的商。提出了两个主要的进一步研究方向。一是研究考克斯环的交换代数性质与变异几何之间的预期相互作用。另一个是光滑环变之间的二元映射的基本分解问题，这将极大地加强Morelli的分解定理。最后提出的方向是森理论的积极特征。Mori理论的绝大部分工作是基于消失定理，它将其限制为特征零。如果这个理论可以推广到正特征，它将有无数的算术应用（例如高维Neron模型的构建）。申请人在这个方向上有显著的结果（通过与之前工作完全不同的方法）。有一种自然的方式，可以使这个结果广泛地普遍化，申请人在这个方向上有部分结果。研究者考虑的主要问题是代数几何中最古老的问题，每个问题至少从20年代开始就被积极研究。正如研究者指出的那样，Mori理论（以其Field获奖发明家命名）的新思想为这些问题提供了新的和有前途的方法。对于曲线模空间的几个基本问题尤其如此，曲线模空间是数学和最近的理论物理中研究最多的对象之一。

The proposed research is in algebraic geometry, covering basic questions in birational geometry related to the areas of Mori theory, moduli of curves, geometric invariant theory, and toric geometry. The first main question is the classical problem of whether or not the moduli space of curves is uniruled. This is a very natural question for Mori theory, which was indeed invented to study uniruledness questions, but somewhat surprisingly no serious attempt from this direction has yet been made --perhaps because the main work in the area was done before the invention of Mori theory. The applicant believes Mori theory gives important new insights, and supports this with many concrete examples. The second proposed direction is on foundational work in birational geometry. The applicant, together with Yi Hu, has observed a foundational connection between Mori theory and geometric invariant theory. They define Mori Dream Spaces abstractly as spaces with the best possible Mori theoretic properties. There turn out to be many examples. They associate to any variety a natural ring, which they call the Cox ring, and show that Mori dream spaces are characterized by the finite generation of this ring, and moreover, are in a natural way a quotient of the associated affine variety in a way generalizing Cox's construction of toric varietie`s as quotients of affine space. There are two main proposed directions for further research. One is to study the expected interplay between commutative algebraic properties of the Cox ring and the geometry of the variety. The other is a foundational factorization problem for birational maps between smooth toric varieties, which would significantly strengthen Morelli's factorization theorem. The final proposed direction is on Mori theory in positive characteristic. The vast majority of work in Mori theory is based on vanishing theorems which restrict it to characteristic zero. The theory would have myriad arithmetic applications (e.g. the construction of higher dimensional Neron models) if it could be extended to positive characteristic. The applicant has a significant result in this direction (by methods entirely different from preceding work). There is a natural way in which this result might vastly generalize, and the applicant has a partial result in this direction.The main questions the investigator considers are among the oldest problems in algebraic geometry, each actively studied at least since the twenties. As the investigator indicates, new ideas from Mori theory (named after its Field's medal winning inventor) suggest new and promising approaches to these problems. This is particularly true of several foundational questions about the moduli space of curves, one of the most studied objects in mathematics and, more recently, theoretical physics.