We prove that the Tate conjecture (and its variants) is invariant under homological projective duality. As an application, we obtain a proof, resp. alternative of in new case linear sections determinantal varieties, old cases Pfaffian cubic fourfolds and complete intersections quadrics. In addition, generalize from schemes to stacks this generalized conjecture(s) for low-dimensional root (twist...

a finite group $g$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. taeri conjectured that an insoluble group satisfying this condition is isomorphic to $s times a$ where $a$ is abelian and $s cong psl_2(q)$ for $q in {4,8}$. we confirm this conjecture.