Global $+$-regularity of regular del Pezzo surfaces in mixed characteristic

TL;DR

This paper proves that certain three-dimensional regular projective schemes over Witt vector rings are globally plus-regular when their special fiber is reduced, extending understanding of regularity in mixed characteristic algebraic geometry.

Contribution

The paper establishes the global plus-regularity of regular del Pezzo surfaces in mixed characteristic under the condition that the special fiber is reduced.

Findings

$M$ is globally $+$-regular if $M_k$ is reduced.

Provides conditions for regularity in mixed characteristic.

Advances the theory of del Pezzo surfaces in algebraic geometry.

Abstract

Let $R = W(k)$ be the ring of Witt vectors over an algebraically closed field $k$ of characteristic $p > 2$. Let $M$ be a three-dimensional regular integral flat projective $R$-scheme such that $H^0(M,\mathcal{O}_M) = R$ and the anticanonical sheaf $\omega_M^{-1}$ is ample. We show that $M$ is globally $+$-regular if the closed fiber $M_k$ is reduced.