The Lüroth problem (Arnaud Beauville, Université de Nice)

05.10.2017 16:15 – 17:15

The Lüroth problem

The Lüroth problem asks whether every field K with C\subset K \subset 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 (Lüroth, 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 Voisin's idea, and how it leads to a number of new results.

PS. The Basic Notions Colloquium will be followed by an aperitif

Lieu

Acacias, Room 17, Colloque

entrée libre