 | Seminario Matematico e Fisico di Milano |
| Arnaud Beauville
Université de Nice
Recent progress on rationality problems
Lunedì 14 Maggio 2018, ore 15:30
Sala di Rappresentanza, Università di Milano, Via C. Saldini 50, Milano |
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Abstract
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We say that an algebraic variety is unirational if it can be parametrized by rational functions, rational if moreover the parametrization can be chosen to be one-to-one. A very classical problem, called nowadays the Luroth problem, asks whether a unirational variety is necessarily rational. This holds for curves (Luroth, 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 3 years ago 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.
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