A fundamental goal of Algebraic Geometry is to classify algebraic varieties up to isomorphism. This is extremely hard, already for surfaces, and open in general. It has become clear that we can only hope for a classification up to birational maps, that is, isomorphisms between dense open sets. Understanding birational maps is therefore a key step towards the classification of algebraic varieties.
For one of the largest families of algebraic varieties, so-called Mori fibre spaces, any birational map between any two of them is composed of special birational maps called Sarkisov links. For surfaces over nice fields, Sarkisov links are well-understood, but little is known about them in dimension three or higher, over any field.
The understanding of Sarkisov links will mean an enormous advance in the study of birational maps and a substantial leap towards a classification of a large family of algebraic varieties.
The very ambitious aim of this project is to describe all Sarkisov links completely in any dimension and in several non-classical settings in terms of base-locus, contracted hypersurfaces and induced rational map on the bases of the implicated Mori fibre spaces. If achieved, it will revolutionize the study of birational maps and provide new exciting tools to determine classes of algebraic varieties in several settings.
In dimension three and higher, already the classification of Sarkisov links over the field of complex numbers is extremely ambitious.
Another very difficult task is to classify Sarkisov links over a field of positive characteristic, as the geometry of algebraic varieties over such fields is even more challenging than it is over the field of complex numbers.
The Minimal Model program, a major active research area in Biratonal Geometry, has made tremendous advances in the last decades. Recently developed ideas and techniques allow the attack on birational maps between algebraic varieties by studying Sarkisov links.
Fields of science
- HORIZON.1.1 - European Research Council (ERC) Main Programme