Mori dream spaces and Q-homology quadrics
Paolo Cascini (Imperial College London), Fabrizio Catanese (Universitaet Bayreuth, KIAS Seoul), Yifan Chen (Beihang University, Beijing), Jong Hae Keum (HCMC, KIAS Seoul)

TL;DR
This paper investigates the properties of fake quadrics, particularly their Mori dream surface status, and explores conditions under which certain surfaces are Mori dream surfaces after reduction modulo primes.
Contribution
It demonstrates that Shavel type surfaces are not Mori dream surfaces but can become so after reduction modulo infinitely many primes, and analyzes which isogenous surfaces are fake quadrics of even type.
Findings
Shavel type surfaces are not Mori dream surfaces.
Infinitely many primes exist where reductions are Mori dream surfaces.
Certain isogenous surfaces are identified as fake quadrics of even type.
Abstract
We show that Shavel type surfaces are fake quadrics of even type which are not Mori dream surfaces, yet there are infinitely many primes such that the reduction modulo is a Mori dream surface. We investigate fake quadrics, first concerning the property of being Mori dream surfaces, then we try to determine which surfaces isogenous to a product are fake quadrics of even type.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Commutative Algebra and Its Applications
