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Fano threefolds in positive characteristic IV

Hiromu Tanaka

TL;DR

<3-5 sentence high-level summary>This paper completes the classification of smooth Fano threefolds in positive characteristic by extending the Mori–Mukai framework to p>0 and systematically analyzing conic bundle structures as the central organizing tool for ρ≥3. It identifies characteristic-two phenomena (wild conic bundles and non-reduced discriminants) that require adjustments to the characteristic-zero list and provides explicit elementary-transform descriptions to connect higher Picard-rank cases to ρ≤2 models. The main contributions include a detailed ρ=3 classification via conic bundles over P^2 and F_1, and a synthesis showing that, up to isomorphism, all ρ=3 Fano threefolds arise from cb structures or from controlled blowups of known ρ≤2 cases, with a complete enumeration of the No. labels tied to the characteristic-zero analogs. The results lay groundwork for Mukai-style descriptions in positive characteristic and clarify how wild phenomena influence the global landscape of Fano threefolds in this setting.

Abstract

Based on the former parts, we classify smooth Fano threefolds of positive characteristic.

Fano threefolds in positive characteristic IV

TL;DR

<3-5 sentence high-level summary>This paper completes the classification of smooth Fano threefolds in positive characteristic by extending the Mori–Mukai framework to p>0 and systematically analyzing conic bundle structures as the central organizing tool for ρ≥3. It identifies characteristic-two phenomena (wild conic bundles and non-reduced discriminants) that require adjustments to the characteristic-zero list and provides explicit elementary-transform descriptions to connect higher Picard-rank cases to ρ≤2 models. The main contributions include a detailed ρ=3 classification via conic bundles over P^2 and F_1, and a synthesis showing that, up to isomorphism, all ρ=3 Fano threefolds arise from cb structures or from controlled blowups of known ρ≤2 cases, with a complete enumeration of the No. labels tied to the characteristic-zero analogs. The results lay groundwork for Mukai-style descriptions in positive characteristic and clarify how wild phenomena influence the global landscape of Fano threefolds in this setting.

Abstract

Based on the former parts, we classify smooth Fano threefolds of positive characteristic.
Paper Structure (36 sections, 139 theorems, 276 equations)

This paper contains 36 sections, 139 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Overviews
  3. Related questions
  4. Mukai's description
  5. Number of deformation families
  6. Preliminaries
  7. Notation
  8. Case of relative Picard number two
  9. Non-Fano blowdowns
  10. Fano threefolds with $\rho=2$
  11. Some non-Fano criteria
  12. Wild conic bundles
  13. Fano conic bundles
  14. Elementary transforms
  15. Reduction to smaller Fano conic bundles
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $X$ be a Fano threefold over $k$, i.e., $X$ is a three-dimensional smooth projective variety over $k$ such that $-K_X$ is ample. Then $X$ is isomorphic to one of threefolds listed in Section s-table. For example, if $\rho(X)=3$

Theorems & Definitions (320)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 310 more