Brauer groups, root stacks, and the Chow ring of the stack of expanded pairs
Algebraic stacks are a type of object studied in algebraic geometry which encode certain geometric moduli problems, in the sense that they keep track of both families of objects and automorphisms of those families. This thesis, consisting of an introduction and three parts, is a collection of works concerning algebraic stacks. The results presented here come in two flavors: firstly, using algebraic stacks as a tool to develop solutions for classical problems, secondly, studying the geometry of
... ng the geometry of algebraic stacks in their own right. The first two parts concern the Brauer group, a classical invariant for fields, which has a generalization to schemes and Deligne-Mumford stacks usingétale cohomology. In the first part, we use root stacks, destackification, and resolution of singularities to generalise an existing result about standard forms of conic bundles to the case of a not necessarily algebraically closed base field. In the second part, we give an interpretation of the residue map from the Brauer group of the quotient field of a discrete valuation ring to the cohomology of the residue field in terms of root stacks and Weil restriction. The given description of the residue map has a geometric application to the residue map for Severi-Brauer bundles in standard form. In the third part, we study a particular example of a moduli stack, namely the stack of expanded pairs T, which was first studied in the context of degeneration formulas in Gromov-Witten theory. We use a logarithmic description of T to calculate its Chow ring, which we find to be the ring of quasisymmetric functions QSym originating in combinatorics. We construct a monoid structure on T, which is shown to induce the comultiplication for the Hopf algebra structure on QSym. Finally, we use an interpretation of T in terms of moduli stacks of curves to obtain the Chow rings of certain moduli stacks of semistable curves. 3 Abstract Algebraische Stacks sind eine Art von Objekt, welches in der algebraischen Geometrie studiert wird. Stacks stellen die Lösung für gewisse Modulprobleme dar, indem sie sowohl Familien von Objekten als auch deren Automorphismen enkodieren. Diese Arbeit, bestehend aus einer Einführung und drei Teilen, ist eine Sammlung von verschiedenen Arbeitenüber algebraische Stacks. Es werden zwei verschiedene Typen von Resultaten präsentiert: erstens der Gebrauch von Stacks als Werkzeug, um Probleme in klassischer algebraischer Geometrie zu lösen, zweitens das Studium der Geometrie von Stacks als Objekt von separatem Interesse. Everyone I had fruitful conversations with or learned from throughout my studies, including, in no particular order: 1 In general, there will be an injection from the Azumaya Brauer group into the cohomological Brauer group, which is an isomorphism e.g. for smooth quasi-projective varieties. Abstract. We generalize a classical result by V. G. Sarkisov about conic bundles to the case of a not necessarily algebraically closed perfect field, using iterated root stacks, destackification, and resolution of singularities. More precisely, we prove that whenever resolution of singularities is available, over a general perfect base field, any conic bundle is birational to a standard conic bundle.