Research in Birational Geometry

拟议的研究是在代数几何，涵盖基本问题，在双有理几何领域的森理论，模量的曲线，几何不变理论，复曲面几何。 第一个主要问题是曲线的模空间是否是uniruled的经典问题。 这是一个非常自然的问题，森理论，这确实是发明研究uniruledness问题，但有点令人惊讶的是，没有认真的尝试，从这个方向尚未作出-也许是因为主要工作在该地区之前，森理论的发明。 申请人认为Mori理论给出了重要的新见解，并以许多具体实例支持这一点。 第二个方向是双有理几何的基础工作。申请人与Yi Hu一起观察到Mori理论与几何不变理论之间的基础联系。 他们将Mori Dream Spaces抽象地定义为具有最佳Mori理论性质的空间。 事实证明有很多例子。他们关联到任何品种的自然环，他们称之为考克斯环，并表明，森梦想空间的特点是有限代这个环，而且，在一个自然的方式商的相关仿射品种的方式推广考克斯的建设环面varietie`s的仿射空间的。提出了两个主要的进一步研究方向。 一个是研究交换代数性质之间的预期相互作用的考克斯环和几何的品种。 另一个是光滑环面簇之间双有理映射的基本因子分解问题，这将大大加强Morelli的因子分解定理。 最后提出的方向是积极性的森氏理论。 绝大多数的工作在森理论是基于消失定理限制它的特征零。 该理论将有无数的算术应用（例如，建设更高的维Neron模型），如果它可以扩展到积极的特点。 申请人在这个方向上有一个重要的结果（通过与以前的工作完全不同的方法）。 有一个自然的方式，这一结果可能会大大推广，和申请人有一个部分结果在这一方向。主要问题的调查员认为是最古老的问题之一，代数几何，每积极研究至少自20年代。 正如研究者所指出的，来自Mori理论（以其领域的获奖发明者命名）的新思想提出了解决这些问题的新的和有前途的方法。 这是特别真实的几个基本问题的模空间的曲线，其中一个最研究的对象，在数学和最近，理论物理。

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.