Research in Birational Geometry

建议的研究是在代数几何中，涵盖二次几何中与Mori理论、曲线的模、几何不变理论和环面几何相关的基本问题。第一个主要问题是曲线的模空间是否未知的经典问题。对于森氏理论来说，这是一个非常自然的问题，它确实是为了研究独特性问题而发明的，但令人惊讶的是，还没有人从这个方向进行认真的尝试--可能是因为这一领域的主要工作是在森氏理论发明之前完成的。申请人认为森友理论提供了重要的新见解，并用许多具体的例子来支持这一点。第二个建议的方向是关于双曲面几何的基础工作。申请人和易虎一起观察到了森氏理论和几何不变量理论之间的基本联系。他们将Mori Dream空间抽象地定义为具有最好的Mori理论性质的空间。事实证明，这样的例子很多。证明了Mori梦空间的特征是这个环的有限生成，而且在某种程度上，Mori梦空间是相关仿射簇的商，从而推广了Cox对环状簇的构造为仿射空间的商。提出了进一步研究的两个主要方向。其一是研究Cox环的交换代数性质与簇的几何之间的相互作用。另一类是光滑环簇之间的双调映射的基本因式分解问题，这将大大加强Morelli的因式分解定理。最后提出的方向是关于森喜朗理论的积极特征。森氏理论的绝大多数工作都是基于将其限制为特征零的消失定理。如果该理论能推广到正特征，它将会有无数的算术应用(例如，构造高维Nelon模型)。申请人在这方面取得了重大成果(通过与以前工作完全不同的方法)。有一种自然的方式可以极大地推广这一结果，申请人在这个方向上有一个部分结果。研究人员考虑的主要问题是代数几何中最古老的问题之一，每个问题至少从二十年代起就被积极研究。正如这位研究人员指出的那样，森友理论(以菲尔德获奖发明家的名字命名)的新想法提出了解决这些问题的新的、有前途的方法。关于曲线的模空间的几个基本问题尤其如此，它是数学和最近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.