  This note is an extended version of the slides for my talk with the same
title at the {\it Arithmetic, geometry, and modular forms: a conference in
honour of Bill Duke} in June 2019 at the ETH in Z"urich. The results presented
concern three geometric criteria for the integrality of factorial ratios,
numbers such as (30n)!n!/(6n)!(10n)!(15n)!, which are integral in a
non-immediate way for all n. This work is an offshoot of an ongoing project on
hypergeometric motives joint with D. Roberts and M. Watkins.
