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Arithmetic and geometric deformations of 3-folds

Fabio Bernasconi, Iacopo Brivio, Stefano Filipazzi

Abstract

We show that mixed-characteristic and equi-characteristic small deformations of 3-dimensional canonical (resp. terminal) singularities with perfect residue field of characteristic $p>5$ are canonical (resp. terminal). We discuss applications to arithmetic and geometric families of 3-dimensional Fano varieties and minimal models with canonical singularities. Our results are contingent upon the existence of log resolutions of 4-folds.

Arithmetic and geometric deformations of 3-folds

Abstract

We show that mixed-characteristic and equi-characteristic small deformations of 3-dimensional canonical (resp. terminal) singularities with perfect residue field of characteristic are canonical (resp. terminal). We discuss applications to arithmetic and geometric families of 3-dimensional Fano varieties and minimal models with canonical singularities. Our results are contingent upon the existence of log resolutions of 4-folds.
Paper Structure (11 sections, 26 theorems, 9 equations)

This paper contains 11 sections, 26 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Singularities, models, and families
  5. Deformations of canonical surface singularities
  6. Running a birational MMP over a DVR in dimension 4
  7. Deformations of 3-folds in positive and mixed characteristic
  8. Deformations of 3-dimensional canonical singularities
  9. Applications to families of Fano varieties and minimal models
  10. Deformations of non-klt canonical singularities in characteristic 0
  11. Jumps of the Cartier index of $K_X$

Key Result

Theorem 1.1

Let $R$ be an excellent DVR with perfect residue field $k$ of characteristic $p>5$. Let $(X, \Delta) \to \mathop{\mathrm{Spec}}\nolimits(R)$ be a family of 3-dimensional couples such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier. If $(X_k, \Delta_k)$ is klt (resp. log canonical), then $(X, X_k+\Delta)$

Theorems & Definitions (63)

  • Theorem 1.1: cf. \ref{['cor: plt-inv-adj']}, \ref{['cor: inv-adjunction']}
  • Theorem 1.2
  • Remark 1
  • Corollary 1: cf. \ref{['cor: def_Fano']}
  • Corollary 2: cf. \ref{['cor: lift_MM_char0']}
  • Theorem 1.3: see \ref{['jump:perfect']}
  • Definition 1
  • Remark 2
  • Definition 2
  • Remark 3
  • ...and 53 more