Birational Geometry

In the period June 8-10, 2020, the International Center for Mathematical Sciences – Sofia at the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences is organizing a workshop on birational geometry. The goal of this workshop is to discuss recent advances in the field.

The workshop is supported by the Simons Collaborative Grant – HMS, NRU HSE, RF government grant, ag. 14.641.31.000, Simons Principle Investigator Grant, Bulgarian National Scientific Program “VIHREN”, Project № KP-06-DV-7.

The main speakers will be Yuri Tschinkel (Courant Institute, New York University and Simons foundation, USA), Ivan Cheltsov (University of Edinburgh, UK), Yu-Wei Fan (University of California, Berkeley) and Ludmil Katzarkov (University of Miami, USA; Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences; National Research University Higher School of Economics, Russian Federation).

They will give lectures on the following topics:

Prof. Yuri Tschinkel: Rational points on algebraic varieties
Abstract: I will discuss the main results and open problems concerning the arithmetic of curves, surfaces, and threefolds. Among the topics: classification schemes, rationality problems, arithmetic invariants, existence and distribution of rational and integral points, analytic techniques in Diophantine geometry.

Prof. Ivan Cheltsov: K-Stability of Fano Varieties
Abstract: A smooth Fano manifold admits a Kähler-Einstein metric if and only if it is K- polystability (K-stability if the automorphism group is finite). The goal of this course is to explain how to prove and disprove K-polystability and K-stability using basic tools of birational geometry. The course will be focused on smooth del Pezzo surfaces (two-dimensional Fano manifolds) and smooth Fano threefolds (three-dimensional Fano manifolds). In two-dimensional cases, the problem is completely solved by Tian. But the problem is very far from being solved for threefolds. We will give a short proof of Tian’s theorem and discuss in details several three-dimensional examples.

Lecture 1: Birational definition of K-stability. Alpha-invariant, beta-invariant, delta-invariant. K-stability of smooth del Pezzo surfaces.

Lecture 2: Computing alpha-invariants of some smooth Fano threefolds to prove K-stability. Using beta-invariant to prove K-instability.

Prof. Yu-Wei Fan: New rational cubic fourfolds arising from Cremona transformations
Abstract: It is conjectured that two cubic fourfolds are birational if their associated K3 categories are equivalent. We prove this conjecture for very general cubic fourfolds of discriminant 20, where the birational maps are produced via certain Cremona transformations defined by Veronese surfaces. Using these birational maps, we find new rational cubic fourfolds.