The lectures will discuss cohomological and Chow-theoretic obstructions to rationality or stable rationality of complex projective varieties. Of course, there are many such geometric obstructions, like the plurigenera, but we will rather focus on the case of rationally connected varieties, where these obvious obstructions are trivial. We will discuss:
This series of lectures will focus on birational rigidity of rational connected varieties, and related topics such as solid Fano varieties. Birationally rigid varieties, among rationally connected ones, sit at the opposite end of the spectrum when compared to rational varieties. Their minimal model end-product is unique, hence the terminology. A smooth quartic 3-fold is the first know example of a birationally rigid variety. The main method used to prove rigidity is the method of maximal singularities, initiated by the celebrated work of Iskovskikh and Manin in the seventies. Other methods, include the use of Sarkisov links, analysis of the Mori cones of blow ups, classification of extremal contraction, and super-maximal singularities. Solid Fanos, introduced by Ahmadinezhad, generalise birationally rigid Fano varieties allowing us to get closer to a sensible classification of Fano 3-folds. All these will be discussed in great depth, with many explicit working examples. We also discuss an equivariant version of the theory. Some applications to the classification of conjugacy classes of subgroups of the Cremona group will also be discussed.