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Семинар Лаборатории алгебраической геометрии и ее приложений
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[Rational curves on K3 surfaces] Кристиан Лидтке Мюнхенский технический университет |
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Аннотация: Although a projective K3 surface over the complex numbers cannot be unirational, Bogomolov conjectured that it contains nevertheless infinitely many rational curves. In this talk, I will prove Bogomolov's conjecture for K3 surfaces of odd Picard rank - this includes the generic case of Picard rank 1. The idea is to reduce to positive characteristic p, and to use the Tate conjecture to find infinitely many rational curves (albeit for infinitely many distinct primes). Then we use Kontsevich's moduli space of stable maps and a rigidification argument to lift cycles of rational curves to characteristic zero, which eventually establishes infinitely many rational curves on the original K3 surface. This work is joint with Jun Li and extends an approach due to Bogomolov, Hassett, and Tschinkel. Язык доклада: английский |
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