Cohomological and Geometric Approaches to Rationality Problems
We prove vanishing results for unramified stable cohomology of finite groups of Lie type.
We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane, and singular cubics, and discuss its relation to the rationality of these cubics.
We study Noether’s rationality problem for actions of finite groups on projective three-space.
We study properties of double covers of ℙ3 ramified along nodal sextic surfaces such as nonrationality, ℚ-factoriality, potential density, and elliptic fibration structures. We also consider some relevant problems over fields of positive characteristic.