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On the degree of Fano threefolds with canonical Gorenstein singularities

Yuri G. Prokhorov

TL;DR

The work establishes a sharp bound $-K_V^3\le 72$ for Fano threefolds with canonical Gorenstein singularities by reducing to weak Fano models via the MMP, analyzing extremal contractions and conic-bundle structures, and treating base dimensions $Z$ of $0$, $1$, or $2$. The core method heightens the role of crepant birational maps and reductions to simpler models (e.g., $\mathbb P^2$- or $\mathbb P^2$-bundle geometries) to bound the anti-canonical degree; equality is achieved only by the weighted projective spaces ${\mathbb P}(3,1,1,1)$ and ${\mathbb P}(6,4,1,1)$. This resolves the Fano-Iskovskikh conjecture in full generality and yields corollaries about hyperplane sections, cone structures, and deformations for varieties with canonical singularities. The results illustrate how extremal-ray analysis and conic-bundle techniques tightly constrain the geometry and degree of Fano threefolds in the canonical setting.

Abstract

We consider Fano threefolds $V$ with canonical Gorenstein singularities. A sharp bound $-K_V^3\le 72$ of the degree is proved.

On the degree of Fano threefolds with canonical Gorenstein singularities

TL;DR

The work establishes a sharp bound for Fano threefolds with canonical Gorenstein singularities by reducing to weak Fano models via the MMP, analyzing extremal contractions and conic-bundle structures, and treating base dimensions of , , or . The core method heightens the role of crepant birational maps and reductions to simpler models (e.g., - or -bundle geometries) to bound the anti-canonical degree; equality is achieved only by the weighted projective spaces and . This resolves the Fano-Iskovskikh conjecture in full generality and yields corollaries about hyperplane sections, cone structures, and deformations for varieties with canonical singularities. The results illustrate how extremal-ray analysis and conic-bundle techniques tightly constrain the geometry and degree of Fano threefolds in the canonical setting.

Abstract

We consider Fano threefolds with canonical Gorenstein singularities. A sharp bound of the degree is proved.

Paper Structure

This paper contains 10 sections, 57 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Notation, conventions and preliminary results
  3. Some facts from the minimal model theory
  4. Contractions of extremal rays on weak Fano threefolds
  5. The conic bundle case
  6. Construction
  7. Case: $Z$ is a point
  8. Case: $X_{\eta}\simeq{\mathbb P}^2$
  9. Case: $X_{\eta}\simeq{\mathbb P}^1\times{\mathbb P}^1$
  10. Case: $\dim Z=2$

Key Result

Theorem 1.1

Let $Y$ be a ${\mathbb Q}$-Fano threefold with ${\mathbb Q}$-factorial terminal singularities and Picard number $1$. If the anti-canonical model $\Phi_{|-K_Y|}(Y)$ is three-dimensional, then $Y$ is birationally equivalent to a Fano threefold $V$ with canonical Gorenstein singularities and base point

Theorems & Definitions (104)

  • Theorem 1.1: A
  • Conjecture 1.2: Fano, Iskovskikh
  • Theorem 1.3: Namikawa
  • Example 1.4: cf. Fano, Isk0
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Conjecture 1.9: cf. Theorem \ref{['th-Namik']}
  • Remark 1.10
  • ...and 94 more