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Regular version of the site

Alexander Duncan

Alexander Duncan (University of Michigan) visited Laboratory of Algebraic Geometry in April 2015 and June 2018.


On June 15th 2018 Alexander Duncan gave a talk
"Exceptional collections on arithmetic toric varieties"  at the seminar of Laboratory.

Abstract:
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.

On April 29th 2015
Alexander Duncan gave a talk
"Equivariant unirationality of surfaces" at the seminar of Laboratory.

Abstract:
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.

 

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