Mori dream spaces and Q-homology quadrics

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 $p$ such that the reduction modulo $p$ 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.