Is the Tate Conjecture stable under birational equivalence? In particular, is the Tate Conjecture for rational varieties known?
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2$\begingroup$ Take an arbitrary smooth projective $X$, embed it in $\mathbf{P}^N$ (with codimension $>1$), and consider $Z = {\rm Bl}_X \mathbf{P}^N$. Then by the formulas for cohomology and Chow ring of a blowup, I think the Tate conjecture for $X$ and $Z$ should be equivalent statements. $\endgroup$– Piotr AchingerCommented Feb 22, 2018 at 9:08
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$\begingroup$ Correct (the comment). In particular it is enough to check the Tate Conjecture for rational varieties. A similar argument works for the Hodge conjecture $\endgroup$– user87684Commented Feb 22, 2018 at 9:12
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