Moduli of Rational Curves with Marked Points and Beyond

这个项目涉及代数几何中的问题，代数几何是现代数学的核心领域之一，与其他领域(从复杂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、系统发育和弦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示。给定类型的代数簇的变化被“模空间”捕获，模空间的点代表这些代数簇。这个项目以模空间为中心，它的点表示稳定的点有理曲线，它们本身就是具有非常丰富结构的代数变体。这些模空间构成了更复杂的模空间的积木，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，与代数族的分类有关，与数论建立新的联系，与物理学的深化联系。最终目标是从三个不同的角度更广泛地理解代数族。这些项目源于围绕模空间M(0，n)的公开问题：它的有效圈是什么，它的算术交理论是什么，它的派生范畴是什么。第一个项目涉及环型变种的爆破的双曲几何。直到最近，人们对森梦空间财产在单一概括点上的环形品种崩盘的失败知之甚少。随着最近的新技术的出现，关于曲面上的线性曲线系统的长期悬而未决的问题有望取得进一步的进展。作者还致力于发展M(0，n)的Arakelov理论，并与双曲几何建立联系。最后，该项目的目的是在M(0，n)及其相关的模空间上构造等变的、完全的、例外的集合。应用包括对一系列其他代数变种的派生范畴的明确理解。

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.