Time: 9h00, Wednesday, March 9, 2016
Location: Room 6, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi
Abstract: The L\"uroth problem asks whether every field $K$ with $\mathbb{C}\subset K \subset \mathbb{C}(x_1,\ldots ,x_n)$ is of the form $\mathbb{C}(y_1,\ldots ,y_p)$. In geometric terms, if an algebraic variety can be parametrized by rational functions, can one find a one-to-one such parametrization?
This holds for curves (L\"uroth, 1875) and for surfaces (Castelnuovo, 1894); after various unsuccessful attempts, it was shown in 1971  that the answer is quite negative in dimension 3: there are many examples of unirational varieties which are not rational. Up to last year the known examples in dimension >3 were quite particular, but a new idea of Claire Voisin has significantly improved the situation.
I will survey the colorful history of the problem, then explain the three different methods found in the 70's to get counter-examples. Then I will discuss Voisin's idea, and how it leads to new examples.