Periodic Reporting for period 1 - Saphidir (SArkisov Program in HIgher Dimension, over Imperfect fields and for birRegulous maps)
Reporting period: 2023-01-01 to 2025-06-30
The objects of study of this project are geometric objects described by polynomial equations, for instance a sphere or an ellipse, and they are called algebraic varieties.
A broad objective of Algebraic Geometry is to classify algebraic varieties up to isomorphisms. This is, however, much too hard, already for surfaces, and one therefore aims to classify them up to so-called birational maps, which are isomorphisms between Zariski-open dense subsets of algebraic varieties. One can view birational maps as an analogue of symmetries, only that one is allowed to remove points or curves.
The Minimal Model Program is a way to simplify algebraic varieties with birational maps, but it remains to classify the outputs up to birational maps between themselves.
Certain types of output varieties have very few birational maps, like varieties of general type, and other types may have plenty of birational maps, like Mori fibre spaces. This project studies birational maps between Mori fibre space, and in general they can be very wild. The Sarkisov program states that any birational map between Mori fibre spaces is a composition of so-called Sarkisov links; these are chunks of birational maps that seem controllable. They are listed in dimension two and in the case of toric threefolds. The aim of the project is to list them reasonably in any dimension.
A second objective of the project is to study birational maps between real algebraic varieties that extend continuously at every real point; they are called biregulous maps. They are hard to study and currently not much is known about the group of biregulous maps of the plane, and even less is known in higher dimension. The aim of the project is to describe properties of the group of biregoulous maps of the projective space of dimension two and higher.
A broad objective of Algebraic Geometry is to classify algebraic varieties up to isomorphisms. This is, however, much too hard, already for surfaces, and one therefore aims to classify them up to so-called birational maps, which are isomorphisms between Zariski-open dense subsets of algebraic varieties. One can view birational maps as an analogue of symmetries, only that one is allowed to remove points or curves.
The Minimal Model Program is a way to simplify algebraic varieties with birational maps, but it remains to classify the outputs up to birational maps between themselves.
Certain types of output varieties have very few birational maps, like varieties of general type, and other types may have plenty of birational maps, like Mori fibre spaces. This project studies birational maps between Mori fibre space, and in general they can be very wild. The Sarkisov program states that any birational map between Mori fibre spaces is a composition of so-called Sarkisov links; these are chunks of birational maps that seem controllable. They are listed in dimension two and in the case of toric threefolds. The aim of the project is to list them reasonably in any dimension.
A second objective of the project is to study birational maps between real algebraic varieties that extend continuously at every real point; they are called biregulous maps. They are hard to study and currently not much is known about the group of biregulous maps of the plane, and even less is known in higher dimension. The aim of the project is to describe properties of the group of biregoulous maps of the projective space of dimension two and higher.
Partial results towards a classification of Sarkisov links has been achieved in several publications in the context of Calabi-Yau pairs of dimension 3 and the volume invariant Sarkisov program (see publication list).
Further results in the context of the equivariant Sarkisov program have been achieved in dimension 3 and over the field of real numbers and for Mori fibre spaces over ruled surfaces.
Results on the Sarkisov program have been applied to show K-stability of certain 3folds.
The project has shown the Sarkisov program for excellent surfaces and has classified rational Mori fibre spaces in dimension 2 over imperfect fields and the Sarkisov linkes between them.
Further results in the context of the equivariant Sarkisov program have been achieved in dimension 3 and over the field of real numbers and for Mori fibre spaces over ruled surfaces.
Results on the Sarkisov program have been applied to show K-stability of certain 3folds.
The project has shown the Sarkisov program for excellent surfaces and has classified rational Mori fibre spaces in dimension 2 over imperfect fields and the Sarkisov linkes between them.
All the above results are beyond the state-of-the art.