Người báo cáo: Prof. A. Beauville, Laboratoire J.-A. Dieudonné Université de Nice, France
Thời gian: 9h-10h, Thứ 4, ngày 9/3/2016
Địa điểm: Phòng số 6, nhà A14, Viện Toán học, 18 Hoàng Quốc Việt, Hà Nội
Tóm tắt: 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. |
