During our stay at Erwin Schrödinger International Institute for Mathematics and Physics, we plan to study G-equivariant birational geometry and rationality problem for three-dimensional rationally connected varieties equipped with action of a finite group G that belongs to a certain class (simple, almost simple, insoluable etc).
As an application, we plan to answer negatively the conjecture posed by Dolgachev in December 2010 during Fifth Iberoamerican Congress on Geometry in Pucon. Namely, we plan to construct a finite group of essential dimension 3 that is not contained in the space Cremona group. During our stay, we also plan to study explicit G-Mori fibre spaces and G-Sarkisov links between them. In particular, we plan to describe all G-Sarkisov links that starts with projective space (of dimension 3) in the case when G is an icosahedral group.
We also aim to describe all G-Mori fibre spaces birational to the projective space in the case when G is an imprimitive group.
We plan to generalize Iskovskikh-Manin-Pukhlikov-Mella theorem about birational rigidity of minimal nodal quartic threefolds for G-minimal ones.
Finally, we plan to complete the work of Corti-Pukhlikov-Ried-Cheltsov-Park by studying rationality problem for Fano 3-fold hypersurfaces of high index.
Research Team: Hamid Ahmadinezhad (Loughborough U), Ivan Cheltsov (U of Edinburgh), Jihuan Park (Pohang U of Science and Technology), Konstantin Shramov (Steklov Institute, Moscow)