Mori Dream Spaces and Rational Curves

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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.