The proposed research is in the field of algebraic geometry. It focuses on a series of problems in classification theory that are connected by the use of similar tools coming from singularity theory and the minimal model program. The project has three main objectives: (1) Investigate abstract properties of numerical invariants of singularities and their impact in birational geometry, and thus address longstanding open problems of primary importance in the field such as Shokurov's ACC Conjectures and the Termination of Flips Conjecture. (2) Determine rigidity properties of the birational geometric structure of certain class of projective Fano hypersurfaces, and use variations of these properties to introduce a new point of view in measuring various degrees of non-rationality. (3) Study different aspects concerning the behavior of curves on Fano varieties, especially in connection to the deformations of the variety and within the context of the minimal model program: expected deformation properties, if satisfied, would bring to light a rather surprising rigidity nature of Fano varieties. Other more speculative projects, notably one regarding a possible connections between Shokurov's ACC Conjecture and resolution of singularities via normalized Nash blow-ups, and one on the behavior of the Cox ring in families of Fano varieties, are outlined throughout the proposal.
The minimal model program, the main program geared towards the birational classification of algebraic varieties, is the general framework of the proposed research. Designed over the model of classification of surfaces (one of the main achievements of the Italian school at the beginning of the 20th century), the minimal model program has become one of the major trends in algebraic geometry, earning S. Mori the Fields Medal for is work on three dimensional varieties. Several leading mathematicians have deeply contributed to the program throughout the years, and very recently there has been some spectacular progress which is bringing us very close to a complete program in all dimensions. There are however many fundamental questions that still remain open, and some of these constitute the main objectives of the proposed research. A series of educational activities are also proposed: the range of such activities covers a large spectrum, from of high school students to peer researchers, passing through undergraduate students, graduate students, and young researchers.