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Mathematics Colloquium


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


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