Arithmetic and geometric deformations of threefolds

We show that mixed‐characteristic and equi‐characteristic small deformations of 3‐dimensional canonical (resp., terminal) singularities with perfect residue field of characteristic p>5$p>5$ are canonical (resp., terminal). We discuss applications to arithmetic and geometric families of 3‐dimensional Fano varieties and minimal models with canonical singularities. Our results are contingent upon the existence of log resolutions for fourfolds.