Mori dream spaces and $\mathbb{Q}$-homology quadrics

Paolo Cascini, Fabrizio Catanese, Yifan Chen and JongHae Keum

Vol. 339 (2025), No. 2, 223–242

DOI: 10.2140/pjm.2025.339.223

Abstract

We show that Shavel-type surfaces are fake $\mathbb{Q}$-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 $\mathbb{Q}$-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 $\mathbb{Q}$-homology quadrics.

Keywords

surfaces of general type, Mori dream space, homology quadric, fake quadric, group action, surface isogenous to a product, foliation

Mathematical Subject Classification

Primary: 14J15, 14J25, 14J29, 14J80

Milestones

Received: 31 January 2025

Revised: 23 June 2025

Accepted: 25 August 2025

Published: 21 September 2025