Perfectoid splitting and global $+$-regularity for smooth hypersurfaces

TL;DR

This paper establishes conditions under which smooth hypersurfaces over certain rings are perfectoid split or globally $+$-regular, depending on the characteristic and degree, advancing understanding in algebraic geometry.

Contribution

It proves that smooth Calabi--Yau hypersurfaces are perfectoid split under certain conditions and that unramified lifts of smooth Fano hypersurfaces are globally $+$-regular, extending geometric regularity results.

Findings

Smooth Calabi--Yau hypersurfaces are perfectoid split if p > dimension and p does not divide d.

Unramified lifts of smooth Fano hypersurfaces are globally $+$-regular if p ≥ dimension and p does not divide d.

Abstract

In this paper, we prove that smooth Calabi--Yau hypersurfaces of degree $d$ over complete unramified discrete valuation rings with residue characteristic $p$ are perfectoid split if $p$ is larger than the relative dimension and $p\nmid d$. We also show that unramified lifts of smooth Fano hypersurfaces over fields of characteristic $p>0$ are globally $+$-regular if $p\ge \dim X$ and $p\nmid d$.