The proposal presents several problems related to the geometry of algebraic varieties and their moduli. A first research objective is to study the birational geometry of the Grothendieck-Knudsen moduli space of stable rational curves, in particular, via arithmetic techniques; a sub-project is to develop the Arakelov theory of this space. A second objective is to prove that 2-Fano manifolds are rationally simply connected. This is an analogue of the celebrated Kollár-Miyaoka-Mori result that Fano manifolds are rationally connected. Using results of de Jong and Starr, a consequence will be a generalization of Tsen's theorem, namely, that a 2-Fano manifold defined over the function field of a surface has a rational point. This would give a natural geometric condition for the existence of rational points.
The broader context of the project is the area of algebraic geometry, 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 given by the solutions of systems of polynomial equations. The variation of algebraic varieties is captured by the so-called moduli spaces, which are themselves algebraic varieties with a very rich structure. In the two main themes of this project, moduli spaces play a central role, both as spaces whose geometry we investigate, and as tools for answering questions about whether certain systems of polynomial equations have solutions or not. The expectation is that the projects will significantly impact other areas of mathematics, especially arithmetic geometry.
