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A canonical Fano threefold has degree $\leq 72$

Chen Jiang, Tianqi Zhang, Yu Zou

TL;DR

The paper proves that the anti-canonical volume $(-K_X)^3$ of a canonical weak Fano $3$-fold satisfies $(-K_X)^3\le 72$, and that this bound is optimal with equality characterized by a crepant birational model to $X_0\simeq \mathbb{P}(1,1,1,3)$ or $\mathbb{P}(1,1,4,6)$. The authors develop two special birational models from a $Q$-factorial terminal weak Fano $3$-fold and apply the minimal model program to relate volumes to del Pezzo and conic bundle structures, ultimately reducing to explicit surface-volume computations. They establish an accompanying bound on $h^0(X,-mK_X)$ and describe equality scenarios very explicitly, including toric crepant extractions of the two singular models. The results confirm Prokhorov's conjecture in the canonical weak Fano setting, and provide a refined birational framework for bounding anti-canonical volumes via Mori fiber spaces and fixed divisors. This has significance for the classification of Fano-type varieties and for understanding sharp volume bounds in higher-dimensional birational geometry.

Abstract

We show that the anti-canonical volume of a canonical weak Fano $3$-fold is at most $72$. This upper bound is optimal.

A canonical Fano threefold has degree $\leq 72$

TL;DR

The paper proves that the anti-canonical volume of a canonical weak Fano -fold satisfies , and that this bound is optimal with equality characterized by a crepant birational model to or . The authors develop two special birational models from a -factorial terminal weak Fano -fold and apply the minimal model program to relate volumes to del Pezzo and conic bundle structures, ultimately reducing to explicit surface-volume computations. They establish an accompanying bound on and describe equality scenarios very explicitly, including toric crepant extractions of the two singular models. The results confirm Prokhorov's conjecture in the canonical weak Fano setting, and provide a refined birational framework for bounding anti-canonical volumes via Mori fiber spaces and fixed divisors. This has significance for the classification of Fano-type varieties and for understanding sharp volume bounds in higher-dimensional birational geometry.

Abstract

We show that the anti-canonical volume of a canonical weak Fano -fold is at most . This upper bound is optimal.

Paper Structure

This paper contains 11 sections, 13 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singularities
  4. Volumes
  5. Ruled surfaces and weak del Pezzo surfaces
  6. Birational models of weak Fano varieties
  7. Mori fiber spaces, varieties of Fano type, and Mori dream spaces
  8. Properties of birational models of weak Fano varieties
  9. Two birational models of a $\mathbb{Q}$-factorial terminal weak Fano $3$-fold
  10. Geometry of weak Fano $3$-folds with $\rho\geq 2$
  11. Proof of the main theorem

Key Result

Theorem 1.1

Let $X$ be a canonical weak Fano $3$-fold. Then the following assertions hold:

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 28 more