Mori Dream Spaces and Rational Curves

该项目旨在了解代数变量及其模的几何的不同方面。主要有两个问题：(1)稳定曲线模空间的有效和充足锥。这一系列的项目集中在稳定有理曲线的Grothendieck-Knudsen模空间上。目的是研究模空间的Mori梦空间结构；特别地，给出了它的双缩的模解释，并给出了它的总坐标环的表示。一个新的观点是将模空间解释为与称为超树的新组合结构相关的可约曲线的Brill-Noether轨迹。(2)利用有理曲线的极小支配族研究了高Fano品种。重点讨论了2-Fano变量的分类和Tsen定理的推广。该项目的更广泛背景是代数几何领域，代数几何是数学中最古老、目前最活跃的分支之一，在整个数学领域有着广泛的应用，并延伸到物理和工程领域。代数几何是对代数变量的研究，代数变量是由多项式方程组的零点定义的几何对象。代数变体的变化是由所谓的模空间捕获的，模空间本身就是具有非常丰富结构的变体。该项目旨在揭示曲线的各种模空间的有趣结构（这是数学和理论物理的许多领域的基础）。该项目涉及算术和计算代数几何，这些领域在编码理论、机器人、计算机视觉、系统发育、统计学等领域的应用越来越广泛。

The project aims at understanding different aspects of the geometry of algebraic varieties and their moduli. There are two main topics:(1) Effective and ample cones of moduli spaces of stable curves. This sequence of projects is focused on the Grothendieck-Knudsen moduli space of stable rational curves. The goal is to investigate the Mori Dream Space structure of the moduli space; in particular, give modular interpretations for its birational contractions and give a presentation for its total coordinate ring. A new point of view is the interpretation of the moduli space as a Brill-Noether locus of a reducible curve associated to new combinatorial structures called hypertrees. (2) A study of higher Fano varieties using minimal dominating families of rational curves. The main focus is on the classification of 2-Fano varieties and generalizations of Tsen's theorem.The broader context of the project is the area of algebraic geometry, one of the oldest and currently one of the most active branches of mathematics, with widespread applications throughout mathematics and reaching into physics and engineering. Algebraic geometry is the study of algebraic varieties, which are geometric objects defined by the zeros of systems of polynomial equations. The variation of algebraic varieties is captured by the so-called moduli spaces, which are themselves varieties with a very rich structure. The project aims at revealing the intriguing structure of various moduli spaces of curves (which are fundamental in many areas of mathematics and in theoretical physics). The project impacts arithmetic and computational algebraic geometry, areas which have increasing applications in coding theory, robotics, computer vision, phylogenetics, statistics, etc.