Destackification and Motivic Classes of Stacks
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
This thesis consists of three articles treating topics in the theory of algebraic stacks. The first two papers deal with motivic invariants. In the first, we show that the class of the classifying stack BPGLn is the inverse of the class of PGLn in the Grothendieck ring of stacks for n ≤ 3. This shows that the multiplicativity relation holds for the universal torsors, although it is known not to hold for torsors ingeneral for the groups PGL2 and PGL3.
In the second paper, we introduce an exponential function which can be viewed as a generalisation of Kapranov's motivic zeta function. We use this to derive a binomial theorem for a power operation defined on the Grothendieck ring of varieties. As an application, we give an explicit expression for the motivic class of a universal quasi-split torus, which generalises a result by Rökaeus.
The last paper treats destackification. We give an algorithm for removing stackiness from smooth, tame stacks with abelian stabilisers by repeatedly applying stacky blow-ups. As applications, we indicate how the result can be used for destackifying general Deligne–Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks.
Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2014. , 23 p.
Algebraic geometry, Algebraic stacks, Destackification, Grothendieck ring, Motive, Torus, Classifying stack
Research subject Mathematics
IdentifiersURN: urn:nbn:se:su:diva-107526ISBN: 978-91-7447-989-8OAI: oai:DiVA.org:su-107526DiVA: diva2:748075
2014-10-24, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Edidin, Dan, Professor
Rydh, David, Assistant Professor
At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Manuscript. Paper 3: Manuscript.2014-10-022014-09-182014-10-30Bibliographically approved
List of papers