| The first part of my thesis, which is joint work with Chenyang Xu, deals with degenerations of rationally connected varieties. Here we show that over a field k of characteristic zero a degeneration of a rationally connected variety always contains a rationally connected subvariety, thus answering affirmatively a question raised by Kollar in [20].; The second part of my thesis is about birational equivalence of products of Brauer Severi surfaces. Let k be a field and {lcub} Ci{rcub}1≤i ≤r and {lcub}Di{rcub} 1≤i≤r be two collections of Brauer Severi surfaces (resp. conics) over k. It is shown that the subgroup generated by C'is in Br(k) is equal to the subgroup generated by D'is in Br(k) ⇔ piC i and piDi are birational ⇒ pi Ci and piDi define the same class in M(k), the Grothendeick ring of varieties over k. The reverse implication of the last arrow holds if char(k) = 0. The proofs are general and some results are valid for arbitrary noetherian base scheme. |
