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Семинар Лаборатории алгебраической геометрии и ее приложений
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[Unirationality, supersingularity, and formal Brauer groups] Кристиан Лидтке Мюнхенский Технический Университет |
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Аннотация: An n-dimensional variety is called unirational if there exists a rational and dominant map from n-dimensional projective space onto it. Unirational varieties are thus (in some sense) close to projective space itself, and the classical Lüroth problem asks whether unirational varieties are in fact rational, that is, birational to projective space. This is true for curves (by Lüroth himself), as well as for surfaces over the complex numbers (Castelnuovo). Over the complex numbers it false in dimension 3 (counter-examples by Fano, Clemens-Griffiths, Artin-Mumford), as well as for surfaces in positive characteristic (Zariski). In this talk I will introduce the formal Brauer group, and various notions of supersingularity (via cycles, formal Brauer groups, and in terms of F-crystals). I will explain that unirational varieties are supersingular. The interesting (and open) question is whether supersingular varieties are unirational. Язык доклада: английский |
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