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Non-quasi-$F$-split canonical affine fourfolds in any characteristic

Teppei Takamatsu, Shou Yoshikawa

TL;DR

This work addresses the construction of canonical Gorenstein affine fourfolds in all positive characteristics that fail to be quasi-$F$-split. The authors construct a family of hypersurfaces $X_m=\mathrm{Spec}\, k[x,y,z,w,t]/(f+t^m)$ using a quartic $f$ with carefully chosen $p$-dependent properties so that $A[t]/(f+t^m)$ is not quasi-$F$-split, and then realize a $ ext{$ ext{Q}$}$-factorial canonical affine fourfold via a controlled sequence of blowups whose exceptional loci are cones over a smooth K3 surface defined by $f$. The proof splits by characteristic, handling $p\neq 2$ and $p=2$ through the $ abla(f)$–Delta$$(f)$$-based obstructions, and relies on Kleiman and KTY results to certify non quasi-$F$-split and $ ext{$ ext{Q}$}$-factoriality. By extending known examples from the $p=3$ case to all positive characteristics, the paper provides explicit, characteristic-wide counterexamples and a Birational framework linking $F$-singularity phenomena to K3 geometry and Artin invariants in dimension four.

Abstract

We construct canonical $\mathbb{Q}$-factorial Gorenstein affine fourfolds in every positive characteristic that are not quasi-$F$-split.

Non-quasi-$F$-split canonical affine fourfolds in any characteristic

TL;DR

This work addresses the construction of canonical Gorenstein affine fourfolds in all positive characteristics that fail to be quasi--split. The authors construct a family of hypersurfaces using a quartic with carefully chosen -dependent properties so that is not quasi--split, and then realize a ext{Q}-factorial canonical affine fourfold via a controlled sequence of blowups whose exceptional loci are cones over a smooth K3 surface defined by . The proof splits by characteristic, handling and through the –Delta-based obstructions, and relies on Kleiman and KTY results to certify non quasi--split and ext{Q}-factoriality. By extending known examples from the case to all positive characteristics, the paper provides explicit, characteristic-wide counterexamples and a Birational framework linking -singularity phenomena to K3 geometry and Artin invariants in dimension four.

Abstract

We construct canonical -factorial Gorenstein affine fourfolds in every positive characteristic that are not quasi--split.
Paper Structure (4 sections, 5 theorems, 49 equations)

This paper contains 4 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Construction
  3. Proof.
  4. Proof.

Key Result

Theorem A

Let $k$ be an algebraically closed field of characteristic $p>0$. Then there exists a $\mathbb{Q}$-factorial canonical Gorenstein affine fourfold $X$ over $k$ that is not quasi-$F$-split.

Theorems & Definitions (17)

  • Theorem A: \ref{['non-qfs-canonical']}
  • Example 1: \ref{['non-qfs-canonical']}, \ref{['2^8-1']}
  • Remark 1
  • Proposition 1
  • proof
  • Claim 1
  • Claim 2
  • Remark 2
  • Remark 3
  • Theorem 2.1: cf. Jang
  • ...and 7 more