FRG: Collaborative research: Birational geometry and singularities in zero and positive characteristic

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The goal of this project is to bring together a group of researchers with experience in a broad array of topics in higher dimensional algebraic geometry, in order to make progress in two closely related areas: birational geometry in positive characteristics and the theory of singularities and linear series arising in the minimal model program. While there has been a lot of recent progress in our understanding of the geometry of higher-dimensional varieties, almost all of this body of work is restricted to characteristic zero, due to the use of vanishing theorems that can fail in positive characteristic. The first goal of this collaborative research is to build tools and a framework that would allow the main results in birational geometry to be extended to positive characteristics. This would make systematic use of the recent techniques that have been devised to exploit the Frobenius morphism. A second goal of the project is to further develop the study of invariants of singularities and of linear series, with an eye toward the remaining problems in the minimal model program. There have been many recent advances in this area. In particular, a conjecture of Shokurov asserting that certain invariants of singularities (the log canonical thresholds) satisfy the so-called ACC property has been solved by some of the PIs. This suggests that other related but harder questions might be within reach, questions whose importance comes from the connection with one of the remaining conjectures in the minimal model program, the termination of flips. The PIs propose to attack one of these problems, predicting the ACC property of another invariant, the minimal log discrepancy. In a separate direction, the PIs plan to undertake a systematic study of examples of linear systems on algebraic varieties that exhibit a pathological behavior from the point of view of various positivity invariants.The last ten years have seen major breakthroughs in the study of higher-dimensional algebraic varieties, but several important problems are still open. A central such problem is intimately related to the study of singularities and one of the goals of this project is to make progress on understanding the properties of the invariants of singularities that appear in this setting. Another general goal of the PIs is to develop systematically the study of algebraic varieties of dimension at least 3 in positive characteristic. A lot less is known in this setting, where new phenomena (sometimes considered pathological) arise. The PIs expect that cross-pollination of ideas with other areas, in particular with commutative algebra, will play an important role in making progress in this direction. Moreover, it is likely that techniques and results in this context would have many applications to other fields (for example, in arithmetic geometry). As part of the collaborative effort, the PIs plan several events that will bring together members of the mathematical community working on related problems and also help disseminate the results and the techniques developed as part of this project.