Publication type: Doctoral thesis
Title: Convex bodies, cox rings and quotients
Authors: Fazlija, Bledar
Advisors / Reviewers: Doran, Brent
Kirwan, Frances
Pandharipande, Rahul
DOI: 10.3929/ethz-b-000254662
Issue Date: 2017
Publisher / Ed. Institution: Universität Zürich
Publisher / Ed. Institution: Zürich
Language: English
Subject (DDC): 500: Natural sciences and mathematics
Abstract: This thesis consists of two chapters that, seemingly distinct at first, are related by the common theme of Okounkov bodies. The main goal of Chapter 2 is to build up a language that enables us, among other things, to approach questions about finite generation of Cox-rings of moduli spaces like M0.n and Mx0,n = M0,n/Sn-1. A recent result suggests that the aforementioned Cox-rings are intriguing tightly connected to the ring Fn = K[(yi - yj) 1≤i<j≤n-1] of translation invariant polynomials. Motivated by the theory of toric varieties, one can analyse Fn in a similar fashion, as one does for a ring K[S], where S is the semigroup of a given toric variety. The structural differences between usual monomials, in the case of toric varieties, and monomials in differences in Fn, is what provides rich structure, encoded in convex and lattice geometry in Chapter 2. Chapter 3, on the other hand, deals primarily with the theme of Okounkov bodies and their relation to GIT quotients. Also, this chapter tries to relate the picture known in the case of toric varieties and toric polytopes to Okounkov bodies on general projective Gvarieties. With the goal of constructing “moment” maps in the setup of Okounkov bodies, projection statements are shown that relate Okounkov bodies on a variety to Okounkov bodies on its GIT quotients. Furthermore, the convex bodies computed in the context of Chapter 2 are shown to be rational polyhedral Okounkov bodies and explicitly described by a set of linear inequalities.
URI: https://digitalcollection.zhaw.ch/handle/11475/18395
License (according to publishing contract): Licence according to publishing contract
Departement: School of Management and Law
Organisational Unit: Institute of Wealth & Asset Management (IWA)
Appears in Collections:Publikationen School of Management and Law

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