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Destackification and Motivic Classes of Stacks
Stockholm University, Faculty of Science, Department of Mathematics.
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.
Keyword [en]
Algebraic geometry, Algebraic stacks, Destackification, Grothendieck ring, Motive, Torus, Classifying stack
National Category
Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-107526ISBN: 978-91-7447-989-8 (print)OAI: oai:DiVA.org:su-107526DiVA: diva2:748075
Public defence
2014-10-24, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript.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.

Available from: 2014-10-02 Created: 2014-09-18 Last updated: 2014-10-30Bibliographically approved
List of papers
1. Motivic classes of some classifying stacks
Open this publication in new window or tab >>Motivic classes of some classifying stacks
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We prove that the class of the classifying stack BPGLn is themultiplicative inverse of the class of the projective linear group PGL_nin the Grothendieck ring of stacks K0(Stack_k) for n = 2 and n = 3 under mild conditions on the base field k. In particular, although it is know that the multiplicativity relation {T} = {S}{PGL_n} does not hold for all PGLn-torsors T -> S, it holds for the universal PGLn-torsors for said n.

National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-107523 (URN)
Available from: 2014-09-18 Created: 2014-09-18 Last updated: 2014-09-18
2. The Binomial Theorem and motivic classes of universal quasi-split tori
Open this publication in new window or tab >>The Binomial Theorem and motivic classes of universal quasi-split tori
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between the Grothendieck groups of these categories. In particular, we derive a binomial formula and use it to give explicit expressions for the classes of universal quasi-split tori in the equivariant Grothendieck group of varieties.

National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-107524 (URN)
Available from: 2014-09-18 Created: 2014-09-18 Last updated: 2014-09-18
3. Functorial destackification of tame stacks with abelian stabilisers
Open this publication in new window or tab >>Functorial destackification of tame stacks with abelian stabilisers
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this article, we study the problem of modifying smooth, algebraic stacks with finite, diagonalisable stabilisers such that their coarse spaces become smooth. The only modifications used are root stacks and blow-ups in smooth centres. If the generic stabiliser of the original stack is trivial, the canonical map from the resulting stack to its coarse space is also a root stack. Hence we can think of the process as removing stackiness from, or destackifying, a smooth stack with help of stacky blow-ups. The construction work over a general base and are functorial in the sense that they respect base change andcompositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how this can be used for destackifying general Deligne-Mumford stacks with finite inertia in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over any field, the method can be used for desingularising locally simplicial toric varieties, without assuming the presence of toroidal structures.

National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-107525 (URN)
Available from: 2014-09-18 Created: 2014-09-18 Last updated: 2014-09-18

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