Moduli of Rational Curves with Marked Points and Beyond

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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.