Germwise singular Fourier--Mukai partnerships

Abstract

We study how local behavior on the base controls whether integral transforms induce fully faithful functors or equivalences between singularity categories of proper flat families. By working with étale germs, we show (under mild hypotheses) that the loci where this happens are open and are compatible with étale base change. Moreover, we give examples, exhibit an autoequivalence of the singularity category which detects regularity of the base, and compute these loci for arithmetic families.