Alexander Duncan
Alexander Duncan (University of Michigan) посетил Лабораторию в апреле 2015 года и в июне 2018 г.
15 июня 2018 г. Alexander Duncan выступил на еженедельном научном семинаре лаборатории с докладом "Exceptional collections on arithmetic toric varieties".
Аннотация:
An "arithmetic toric variety" is a normal variety with a faithful action of an algebraic torus having a dense open orbit. When the base field is algebraically closed, there is only one torus in every dimension and one can identify the torus with its orbit. Over a general field, there may be many non-isomorphic tori of the same dimension. Moreover, it is no longer possible to identify the torus with its orbit since there may not exist any rational points.
Exceptional collections are one way of describing the bounded derived categories of coherent sheaves on a variety. The existence of exceptional collection is a very strong condition but, nevertheless, Kawamata showed that all smooth projective toric varieties possess exceptional collections when the ground field is algebraically closed. For a general field, this immediately fails even in dimension 1. However, if one allows an arithmetic generalization of the "usual" notion of exceptional object, then the theory is again interesting.
I will given an overview of arithmetic toric varieties emphasizing their differences with the algebraically closed case. Then I will discuss exceptional collections over general fields. I will point out the difficulties with generalizing Kawamata's proof as well as some positive results for arithmetic toric varieties.
29 апреля 2015 г. Alexander Duncan выступил с докладом "Equivariant unirationality of surfaces" на еженедельном семинаре Лаборатории
Аннотация:
15 июня 2018 г. Alexander Duncan выступил на еженедельном научном семинаре лаборатории с докладом "Exceptional collections on arithmetic toric varieties".
Аннотация:
An "arithmetic toric variety" is a normal variety with a faithful action of an algebraic torus having a dense open orbit. When the base field is algebraically closed, there is only one torus in every dimension and one can identify the torus with its orbit. Over a general field, there may be many non-isomorphic tori of the same dimension. Moreover, it is no longer possible to identify the torus with its orbit since there may not exist any rational points.
Exceptional collections are one way of describing the bounded derived categories of coherent sheaves on a variety. The existence of exceptional collection is a very strong condition but, nevertheless, Kawamata showed that all smooth projective toric varieties possess exceptional collections when the ground field is algebraically closed. For a general field, this immediately fails even in dimension 1. However, if one allows an arithmetic generalization of the "usual" notion of exceptional object, then the theory is again interesting.
I will given an overview of arithmetic toric varieties emphasizing their differences with the algebraically closed case. Then I will discuss exceptional collections over general fields. I will point out the difficulties with generalizing Kawamata's proof as well as some positive results for arithmetic toric varieties.
29 апреля 2015 г. Alexander Duncan выступил с докладом "Equivariant unirationality of surfaces" на еженедельном семинаре Лаборатории
Аннотация:
A variety X is unirational if there exists a dominant rational map from V to X where V is an affine space. Generalizing this idea, for a linear algebraic group G we say a G-variety X is G-unirational if there exists a G-equivariant dominant rational map from V to X where V is a linear representation. I consider the G-unirationality of del Pezzo surfaces over the complex numbers. I will discuss the connections between the equivariant version ofunirationality and the arithmetic version of unirationality over a non-algebraically closed field.
Анонс доклада
Нашли опечатку?
Выделите её, нажмите Ctrl+Enter и отправьте нам уведомление. Спасибо за участие!
Сервис предназначен только для отправки сообщений об орфографических и пунктуационных ошибках.