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

Abstract We study the plus-pure threshold ( $${{\,\textrm{ppt}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>ppt</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> ) of hypersurfaces in mixed characteristic. We show that the $${{\,\textrm{ppt}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>ppt</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> 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’ $${{\,\textrm{ppt}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>ppt</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> 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 $$p^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> (or modulo other values), bounding their $${{\,\textrm{ppt}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>ppt</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> s. In particular, given a complete unramified regular local ring of mixed characteristic $$p&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , $$f^p + p^2 g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> does not define a perfectoid pure singularity for any f and g . Finally, we compute bounds on the $${{\,\textrm{ppt}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>ppt</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> of hypersurfaces related to elliptic curves. This gives examples where the $${{\,\textrm{ppt}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>ppt</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> is neither the corresponding $${{\,\textrm{fpt}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>fpt</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> in characteristic $$p &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> nor the $${{\,\textrm{lct}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>lct</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> in characteristic zero. This also provides examples where p times the $${{\,\textrm{ppt}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>ppt</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> is not a jumping number, in stark contrast with the characteristic $$p &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math>