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Quasi-$F$-splitting and smooth weak del Pezzo surfaces in mixed characteristic

Hirotaka Onuki, Teppei Takamatsu, Shou Yoshikawa

TL;DR

This work develops a global theory of quasi-$F$-splitting in mixed characteristic by extending Witt-vector techniques and defining Witt divisorial sheaves on normal schemes. A central result is a Kodaira-type vanishing for quasi-$F$-split and Cohen–Macaulay lifts $X$ of projective varieties, enabling vanishing statements for $H^j(X,O_X(L))$ when $A=L-K_X$ is ample and certain CM conditions hold. The authors then connect geometry to algebra via a cone correspondence, linking quasi-$F$-splitting of $X$ to that of its homogeneous coordinate ring, and apply the framework to RDP del Pezzo surfaces, proving lifts are quasi-$F$-split and obtaining vanishing results for liftable divisors. These results show that certain pathologies in positive characteristic disappear after lifting to mixed characteristic, with concrete implications for weak del Pezzo surfaces and related vanishing theorems in mixed and positive characteristics.

Abstract

We introduce the notion of quasi-$F$-splitting in mixed characteristic and study Kodaira-type vanishing on quasi-$F$-splitting varieties. As an application, we prove a Kodaira-type vanishing on lifts of rational double point (RDP) del Pezzo surfaces.

Quasi-$F$-splitting and smooth weak del Pezzo surfaces in mixed characteristic

TL;DR

This work develops a global theory of quasi--splitting in mixed characteristic by extending Witt-vector techniques and defining Witt divisorial sheaves on normal schemes. A central result is a Kodaira-type vanishing for quasi--split and Cohen–Macaulay lifts of projective varieties, enabling vanishing statements for when is ample and certain CM conditions hold. The authors then connect geometry to algebra via a cone correspondence, linking quasi--splitting of to that of its homogeneous coordinate ring, and apply the framework to RDP del Pezzo surfaces, proving lifts are quasi--split and obtaining vanishing results for liftable divisors. These results show that certain pathologies in positive characteristic disappear after lifting to mixed characteristic, with concrete implications for weak del Pezzo surfaces and related vanishing theorems in mixed and positive characteristics.

Abstract

We introduce the notion of quasi--splitting in mixed characteristic and study Kodaira-type vanishing on quasi--splitting varieties. As an application, we prove a Kodaira-type vanishing on lifts of rational double point (RDP) del Pezzo surfaces.
Paper Structure (11 sections, 24 theorems, 85 equations)

This paper contains 11 sections, 24 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The ring of Witt vectors
  4. Witt divisorial sheaves in mixed characteristic
  5. Foundations of quasi-F-splitting
  6. Definition and basic properties of quasi-$F$-splitting
  7. Kawamata--Viehweg-type vanishing
  8. A cone correspondence for quasi-$F$-splitting
  9. RDP del Pezzo surfaces
  10. Quasi-$F$-splitting for explicit hypersurfaces
  11. Applications to weak del Pezzo surfaces

Key Result

Theorem A

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $\overline{X}$ be a normal projective variety over $k$ and $X$ a lift of $\overline{X}$ over $W(k)$. Assume that $X$ is quasi-$F$-split and $X$ is Cohen-Macaulay. Let $L$ be a $\mathbb{Q}$-Cartier Weil divisor such that $A:=L-K_X$

Theorems & Definitions (53)

  • Theorem A: cf. \ref{['thm:KV-vanishing']}
  • Theorem B: \ref{['RDP-dP']}
  • Theorem C: \ref{['van-CM-ample']}
  • Theorem D: \ref{['vanishing']}, cf. KN22*Theorem 1.7 (3)
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • ...and 43 more