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Fano threefolds in positive characteristic I

Hiromu Tanaka

TL;DR

This work extends the Iskovskih–Mori–Mukai program to positive characteristic by classifying smooth Fano threefolds with ρ(X)=1 whose anti-canonical system is not very ample and proves that, when |-K_X| is very ample and the genus satisfies g ≥ 5, the image φ_{|-K_X|}(X) is cut out by quadrics. The authors develop a robust framework built on generic elephants, K3-like surfaces, and an intersection-of-quadrics analysis, adapting Bertini-type results and Δ-genus techniques to positive characteristic. Key contributions include establishing that, in the not-very-ample case, |-K_X| is base point free and yields a double cover onto a low-degree image, and showing that anti-canonically embedded Fano threefolds with g ≥ 5 and ρ=1 are intersections of quadrics, via a refined study of the ambient quadric locus W and its singularities. Collectively, the paper advances the classification of Fano threefolds in characteristic p and provides foundational tools (K3-like vanishing, generic elephants) that may inform future positive-characteristic birational geometry and embedding problems.

Abstract

Over an algebraically closed field of positive characteristic, we classify smooth Fano threefolds of Picard number one whose anti-canonical linear systems are not very ample. Furthermore, we also prove that an anti-canonically embedded Fano threefold of genus at least five is an intersection of quadrics.

Fano threefolds in positive characteristic I

TL;DR

This work extends the Iskovskih–Mori–Mukai program to positive characteristic by classifying smooth Fano threefolds with ρ(X)=1 whose anti-canonical system is not very ample and proves that, when |-K_X| is very ample and the genus satisfies g ≥ 5, the image φ_{|-K_X|}(X) is cut out by quadrics. The authors develop a robust framework built on generic elephants, K3-like surfaces, and an intersection-of-quadrics analysis, adapting Bertini-type results and Δ-genus techniques to positive characteristic. Key contributions include establishing that, in the not-very-ample case, |-K_X| is base point free and yields a double cover onto a low-degree image, and showing that anti-canonically embedded Fano threefolds with g ≥ 5 and ρ=1 are intersections of quadrics, via a refined study of the ambient quadric locus W and its singularities. Collectively, the paper advances the classification of Fano threefolds in characteristic p and provides foundational tools (K3-like vanishing, generic elephants) that may inform future positive-characteristic birational geometry and embedding problems.

Abstract

Over an algebraically closed field of positive characteristic, we classify smooth Fano threefolds of Picard number one whose anti-canonical linear systems are not very ample. Furthermore, we also prove that an anti-canonically embedded Fano threefold of genus at least five is an intersection of quadrics.
Paper Structure (32 sections, 65 theorems, 263 equations)

This paper contains 32 sections, 65 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Overview of the proofs and contents
  3. Generic elephants
  4. K3-like surfaces
  5. Intersection of quadrics
  6. Preliminaries
  7. Notation
  8. Fano threefolds
  9. Canonical surfaces
  10. Cohomologies of Fano threefolds
  11. Bertini theorems
  12. Generic members
  13. General members
  14. $\Delta$-genera
  15. The case of index $\geq 2$
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $k$ be an algberaically 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. Let $r_X$ be the index of $X$ (for its definition, see Definition d-Fano). Assume that $\rho(X)=1$ induced by $|-K_X|$ is a double cover onto its image $Y := \varphi_{|-K_X|}(X)$, i.e., the induced

Theorems & Definitions (162)

  • Theorem 1.1: Theorem \ref{['t-non-bpf-main']}, Section \ref{['s-bpf-case']}
  • Theorem 1.2: Theorem \ref{['t-int-of-quad']}
  • Theorem 1.3: Corollary \ref{['c-generic-ele']}
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 152 more