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Abstract: We show that Shavel-type surfaces are fake Q-homology quadrics of even type which are not Mori dream surfaces, yet there are infinitely many primes p such that the reduction modulo p is a Mori dream surface. We investigate fake Q-homology quadrics, first concerning the property of being a Mori dream surface, then trying to determine which families are of even type among the surfaces isogenous to a higher product which are fake Q-homology quadrics.

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