Knörrer type equivalences for cyclic quotient surface singularities
Friday, 10.5.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Continuous K-theory and cohomology of rigid spaces
Friday, 17.5.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Continuous K-theory is a variant of algebraic K-theory for rigid spaces (nonarchimedean analytic spaces). In this talk, I will relate the bottom K-theory group of a rigid space with its top cohomology group with integral coefficients. I will begin with some recollections about algebraic K-theory for schemes and then introduce continuous K-theory for rigid spaces, as defined by Morrow and further studied by Kerz-Saito-Tamme. Afterwards, I will present an easy proof of my result in the regular case assuming resolution of singularities. This will be done in terms of Berkovich spaces and their skeleta (which will be used as a black box). The general result avoids the assumption of resolution of singularities and works with Zariski-Riemann type spaces instead which are defined as the limit over all models. Despite not a scheme anymore, these Zariski-Riemann type spaces behave, due to a result by Kerz-Strunk, from the K-theoretic point of view similar as a regular model does. The content of this talk is based on my PhD thesis advised by Moritz Kerz and Georg Tamme.
Grothendieck groups of isolated quotient singularities
Friday, 24.5.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
We study the Grothendieck group of the Buchweitz-Orlov singularity category for quasi-projective algebraic schemes. Particularly, we show for isolated quotient singularities that the Grothendieck group of its singularity category is finite torsion and that rational Poincare duality is satisfied on the level of Grothendieck groups. We consider also consequences for the resolution of singularities of such quotient singularities. More concretely we prove a conjecture of Bondal and Orlov on the derived category of rational singularities in the case of quotient singularities.
Gorensteinness and iteration of Cox rings
Friday, 31.5.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
We show that finitely generated Cox rings have trivial canonical class. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones, iteration of Cox rings is finite with factorial master Cox ring.\n