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Birational and non-archimedean geometries

Periodic Reporting for period 2 - BirNonArchGeom (Birational and non-archimedean geometries)

Reporting period: 2019-11-01 to 2021-04-30

Smooth geometric shapes (manifolds) are extensively studied and used in mathematic, physics, computer science, etc. Often a geometric shape (variety), for example, the one given by an equation f(x,y,z) in the three-dimensional space, is singular: it may contain self-intersections, cusps, pinch points, etc. Resolution of singularities is a classical branch of algebraic geometry which studies how a variety can be modified to a smooth manifold, and such modifications are a very useful tool for working with general varieties.

First resolution of singularities over fields containing the rational numbers (the characteristic zero case) was obtained by Hironaka in 1964 and awarded him a Fields medal. Until very recently, Hironaka's method was polished and improved but mathematicians have known essentially a unique resolution algorithm.

The first half of this project addresses the characteristic zero case and its goal is to obtain new methods, which are faster, use no memory, also apply to maps between manifolds, take into account logarithmic structures, etc. This provides new results even over the field of complex numbers and might have applications also outside of mathematics.

The second half of the project plans to study the arithmetic cases of positive and mixed characteristic. After extensive work of many mathematicians only the cases of dimensions 1,2 and 3 were solved so far. I hope that the new tools designed in the first half of the project will be useful here too. These goals of this project are very important for various fields of mathematics, such as arithmetical geometry.
In a series of papers of the PI with D. Abramovich and J. Wlodarczyk (available on archive and submitted for publication) we developed two new methods in the characteristic zero case, which also apply to morphisms and perform faster than the classical algorithm. We expect not only the results, but also the methods (the use of the so-called orbifolds and orbifold weighted blow ups) to become an important tool of birational algebraic geometry and related fields.

In a paper of the PI with Adiprasito and Liu (available on archive and submitted for publication) we established the best possible combinatorial resolution of maps, proving in characteristic zero the semistable conjecture formulated in 2000 by Abramovich and Karu.
Both the new algorithms developed with Abramovich and Wlodarczyk and the semistable conjecture solved with Adiprasito and Liu have taken the field of resolution beyond what was known in the characteristic zero case and introduced essentially new techniques.

The positive characteristic case is even more challenging and many mathematicians think it is out of reach. I expect that the new methods will make it possible to get a partial progress. For example, getting an orbifold resolution (with wild inertia) seems reasonable. I also expect a progress in low dimensional cases, such as resolution of families of surfaces.