The Brauer group of a variety is an important invariant. When the variety is proper and normal, the existence of non-trivial elements in the Brauer group is an obstruction to stable rationality, and it has been used to construct examples of non-rational varieties. In the case of moduli space of sheaves is often related to the existence of the universal family. In this talk, we will show that the Brauer group of the universal moduli stack of vector bundles on (possibly marked) smooth curves of genus at least four over the complex numbers is trivial. As consequence, we obtain an explicit description of the Brauer group of the smooth locus of the associated moduli space of semistable vector bundles. This is a joint work with Roberto Pirisi. |

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