Moduli of Rational Curves with Marked Points and Beyond

该项目涉及代数几何问题，这是现代数学的中心领域之一，与其他领域（从复分析和拓扑学到数论）有多种联系，并在编码理论，计算机代数，遗传学和弦理论等领域有重要应用。代数几何的基本研究对象是代数簇，即多项式方程组解的几何表现。给定类型的代数簇的变化被“模空间”捕获，其点代表这些代数簇。这个项目是集中在模空间，其点代表稳定的尖有理曲线，本身具有非常丰富的结构代数品种。这些模空间形成了更复杂的模空间的构建块，这些模空间在理论物理中起着关键作用。该项目有三个不同的主题，涉及代数簇的分类，与数论建立新的联系，并加深与物理学的联系。最终目标是从三个不同的角度更广泛地理解代数簇。这些项目源于围绕模空间M（0，n）的公开问题：什么是它的有效圈，什么是它的算术交理论，什么是它的派生范畴。第一个项目涉及复曲面品种爆破的双有理几何。直到最近，很少有人知道Mori Dream Space属性在单个一般点上对复曲面品种的爆破失败。随着最近的新技术，有希望对长期悬而未决的问题，在曲面上的曲线的线性系统的进一步进展。研究者还旨在发展M（0，n）的Arakelov理论，并与双有理几何建立联系。最后，该项目的目标是在M（0，n）和相关的模空间上构造等变的、完整的、例外的集合。应用程序包括一个明确的理解派生类别的范围内的其他代数簇。

This project concerns questions in algebraic geometry, one of the central areas in modern mathematics, with multiple connections to other areas (from complex analysis and topology to number theory) and with important applications in fields as diverse as coding theory, computer algebra, phylogenetics, and string theory. The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. The variation of algebraic varieties of a given type is captured by "moduli spaces," whose points represent these algebraic varieties. This project is centered on moduli spaces whose points represent stable pointed rational curves, themselves algebraic varieties with a very rich structure. These moduli spaces form building blocks for more complex moduli spaces that play a key role in theoretical physics. The project has three different themes, related to the classification of algebraic varieties, establishing new connections with number theory, and deepening connections with physics. The ultimate goal is a broader understanding of algebraic varieties from three different perspectives. The projects stem from open questions surrounding the moduli space M(0,n): what are its effective cycles, what is its arithmetic intersection theory, and what is its derived category. A first project concerns the birational geometry of blow-ups of toric varieties. Until recently, very little was known about the failure of the Mori Dream Space property for blow-ups of toric varieties at a single general point. With recent new techniques, there is hope for further progress towards longstanding open questions on linear systems of curves on surfaces. The investigator also aims to develop the Arakelov theory of M(0,n) and establish a connection with birational geometry. Finally, the project aims to construct equivariant, full, exceptional collections on M(0,n) and related moduli spaces. Applications include an explicit understanding of the derived category of a range of other algebraic varieties.