Department of MathematicsRecall that an algebraic variety is rational if it is birational to a projective space. In the past decade it was established that several classes of complex varieties are not (stably) rational: this includes very general hypersurfaces in a certain degree range, cyclic covers, and other varieties.
These results were obtained using the specialization method: to summarize, X is not stably rational if it degenerates to a well chosen mildly singular "reference" variety with some particular nontrivial invariants.
Constructing such reference varieties could be a difficult task in general. In this talk, we will give an overview of the recent progress and we will discuss one specific example of a reference variety, namely, a fibration in cubic surfaces: we will describe a general formula for the invariant coming from the unramified Brauer group.