Birational geometry is a major branch of algebraic geometry, focused on the study of function fields of algebraic varieties. One of its main classical problems, going back to Italian algebraic geometry, is to determine whether or not an algebraic variety \(X\) of dimension \(d\) over a field \(k\) is rational, i.e., birational to projective space \(\mathbb P^d\), or equivalently, whether or not its function field \(F=k(X)\) is a purely transcendental extension over \(k\). Several related notions play an important role:
The three notions coincide for surfaces over algebraically closed fields, but differ over nonclosed fields and in higher dimensions. In addition to being of intrinsic interest, rationality and unirationality properties have various applications. For example:
However, what makes the rationality problem especially appealing is that it has stimulated, and continues to stimulate, major advances in algebraic geometry and number theory. The most notable examples of such are the breakthrough results from 1974, concerning nonrationality of certain 3-folds:
A new geometric method, via degenerations, was introduced by Kollár in 1995: he proved that very general hypersurfaces \(X_d\subset \mathbb P^{n_1}\) of degree \(d\ge 2(n+3)/3\) are not rational.
Seminal ideas on stable rationality, based on the method of specialization and decomposition of the diagonal, were recently introduced by Voisin and Colliot-Thélène--Pirutka. This is the most important achievement in this field during the last 20 to 30 years. It was followed by applications in the works of Beauville, Totaro, and Hassett-Kresch-Tschinkel.
At the same time, we do not yet have answers to some very basic questions. For example: Are cubic hypersurfaces of dimension \(\ge 3\) stably rational? Are nonrational cubic fourfolds dense in moduli?
Recent developments in Homological Mirror Symmetry and derived categories lead to the possibility of using derived categories in order to study the question of rationality. We would like to mention here the results obtained by Bondal, Orlov, Kuznetsov, Thomas, Huybrechts, Addington, Macri, Boehning, Auel, Bernardara, and many others.
This two month long program, which is focused on these recent developments, will help consolidate results that have been obtained and will help find new directions of collaboration.
The Program will contain three workshops: