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If a variety X over a number field k has points in each completion of k, then we can ask whether the set of k-rational points is nonempty. To tackle this question, we can impose conditions on the collection of local points to narrow down the subset that contains the rational points, should any exist. One fruitful approach uses the Brauer group of X, which defines an obstruction set known as the Brauer-Manin set.

Brauer group. It is thus sensible to ask whether properties of this finite subset can be determined in advance, i.e., without computing the Brauer-Manin set. In the case of cubic surfaces, for example, it is known that just one Brauer class is needed to detect an obstruction. In this talk, we'll discuss new results which show that in general we cannot hope to give such quantitative bounds; we construct conic bundles over the projective line for which the Brauer group modulo constants is generated by N classes and all N generators are required to witness an obstruction. (This is joint work with Pagano, Poonen, Stoll, Triantafillou, Viray, Vogt.) 

Link to slides: https://jensberg047.github.io/manybrauer/