Not to alarm you, but if your grandparents were arrested, they could have been exiled to Siberia, sentenced to a labor camp, or even executed.
An uncle told me years ago that my grandfather's brother was arrested around that time, and it was assumed he was deported to Siberia. But I later found my great-uncle's name on a website listing "victims of Stalin," indicating that he was executed (shot) in 1937. I've since learned the complete story from his granddaughter.
There are many books about Stalin's purges during this time. The "Great Purge" occurred in the late 1930s, but there were thousands of other executions carried out under Stalin until his death in 1953.
Researching WEISSMAN/VAYSMAN (Ostropol, Ukraine); MOROZ and ESTRIN (Shklov, Belarus); LESSER/LESZEROVITZ, MAIMAN, and BARNETT/BEINHART/BERNHART (Lithuania/Latvia); and ROSENSWEIG/ROSENZWEIG, KIRSCHEN, and SCHWARTZ (Botosani, Romania)