Let and be schemes of finite type over and let be a finite map. We show the following holds for all sufficiently large primes : If and are any splittings on and , such that the restriction of is compatible with and , and is any compatibly split subvariety of , then the reduction is a compatibly split subvariety of . This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.
"Frobenius split subvarieties pull back in almost all characteristics." J. Commut. Algebra 12 (4) 573 - 579, Winter 2020. https://doi.org/10.1216/jca.2020.12.573