Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings

Abstract We study the plus-pure threshold ( $${{\,\textrm{ppt}\,}}$$ ppt ) of hypersurfaces in mixed characteristic. We show that the $${{\,\textrm{ppt}\,}}$$ ppt limits to the F -pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic extremal singularities cannot attain the same ‘extremal’ $${{\,\textrm{ppt}\,}}$$ ppt values in the unramified setting. We also study equations which have controlled ramification when we adjoin their p -th roots as well as equations which admit p -th roots modulo $$p^2$$ p 2 (or modulo other values), bounding their $${{\,\textrm{ppt}\,}}$$ ppt s. In particular, given a complete unramified regular local ring of mixed characteristic $$p>0$$ p > 0 , $$f^p + p^2 g$$ f p + p 2 g does not define a perfectoid pure singularity for any f and g . Finally, we compute bounds on the $${{\,\textrm{ppt}\,}}$$ ppt of hypersurfaces related to elliptic curves. This gives examples where the $${{\,\textrm{ppt}\,}}$$ ppt is neither the corresponding $${{\,\textrm{fpt}\,}}$$ fpt in characteristic $$p > 0$$ p > 0 nor the $${{\,\textrm{lct}\,}}$$ lct in characteristic zero. This also provides examples where p times the $${{\,\textrm{ppt}\,}}$$ ppt is not a jumping number, in stark contrast with the characteristic $$p > 0$$ p > 0