We show that the Hopf algebra of quasi-symmetric functions arises naturally as the integral Chow ring of the algebraic stack of expanded pairs originally described by J. Li, using a more combinatorial description in terms of configurations of line bundles. In particular, we exhibit a gluing map which gives rise to the comultiplication. We then apply the result to calculate the Chow rings of certain stacks of semistable curves.Notation. Throughout this note, we let k be an algebraically closed field of characteristic 0. There is no loss for the reader in assuming k to be C throughout. We let C be the category of finite ordered sets and order-preserving injections, and we let C ≤m be the full subcategory consisting of sets with m or less elements.

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.

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We reinterpret the residue map for the Brauer group of a smooth variety using a root stack construction and Weil restriction for algebraic stacks, and apply the result to find a geometric representative of the residue of the Brauer class associated to a conic bundle.