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

TL;DR

The paper proves that a regular integral flat projective 3-fold $M$ over the Witt vectors $R=W(k)$ with $H^0(M,\mathcal{O}_M)=R$ and ample $\omega_M^{-1}$ is globally $+$-regular when the closed fiber $M_k$ is reduced (for $p>2$). The proof blends mixed-characteristic quasi-$F$-splitting with a detailed classification of $M_k$ (RDP, normal non-RDP, and nonnormal fibers), using results of Onuki–Takamatsu–Yoshikawa and Reid to establish quasi-$F$-splitting and then deduce global $+$-regularity via a normalization-based criterion; this yields Kawamata–Viehweg-type vanishing for big semiample line bundles on $M$. The work further develops the deformation theory of low-dimensional Fano schemes in mixed characteristic, classifies regular Fano curves and smooth weak del Pezzo surfaces, and provides a comprehensive picture of when del Pezzo fibers lift and are globally $F$-split. Collectively, these results reinforce the conjectural link between global $+$-regularity and Fano-type behavior in mixed characteristic and supply vanishing theorems relevant to the minimal model program in this setting.

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 $ω_M^{-1}$ is ample. We show that $M$ is globally $+$-regular if the closed fiber $M_k$ is reduced.

Theorems & Definitions (87)

• Theorem A
• Theorem B
• Definition 2.1: B+MMPTakamatsuYoshikawa23
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4
• Theorem 2.5: HidakaWatanabe81
• proof
• Theorem 2.7: Reid94