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What are some nontrivial examples of schemes whose Brauer group is trivial? I am aware that rational varieties have trivial Brauer group and retract rational varieties in the sense of Saltman also do. Can someone give me an example of a variety outside this class whose Brauer group is trivial?

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There are plenty! A smooth complex projective variety with $H^2(X,\mathcal{O}_X)=0$ and $H^3(X,\mathbb{Z})$ torsion free has trivial Brauer group. For instance any complete intersection in $\mathbb{P}^n$ of dimension $\geq 3$ will have trivial Brauer group -- and many others.

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Every curve over an algebraically closed field has trivial Brauer group, by Tsen's Theorem.

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And likewise for every (smooth proper) curve over a finite field by class field theory (though it seems that the OP is only asking about varieties over an algebraically closed field). – user76758 Feb 4 '14 at 16:47

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