|Date: Tuesday, April 12, 2016
Title: Special AG lecture series in Spring: The L\"uroth problem
Abstract: The L\"uroth problem asks whether every field K with C < K < C(x_1, ... ,x_n) is of the form C(y_1, ... ,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, three different counter-examples were found in 1971. I will survey the colorful history of the subject, then describe the three methods, and explain that they give a rather satisfactory answer in dimension 3, but only very particular examples in higher dimension. Then I will discuss a new idea of Claire Voisin which has significantly improved the situation.
[This is the first lecture in the "Spring Lectures" series.]
Speaker: Arnaud Beauville
Institution: Universite de Nice