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An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$-Fano threefolds

Chen Jiang, Yu Zou

Abstract

We show that the anti-canonical volume of a canonical weak $\mathbb{Q}$-Fano $3$-fold is at most $324$.

An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$-Fano threefolds

Abstract

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

Paper Structure

This paper contains 13 sections, 12 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singularities of pairs
  4. Non-klt centers
  5. Log canonical thresholds
  6. Volumes
  7. Intersection multiplicities on smooth surfaces
  8. Log canonical thresholds on weak del Pezzo surfaces
  9. Proof of the main theorem
  10. A reduction step
  11. Weak del Pezzo fibration case
  12. Conic bundle case
  13. Proofs of Theorem \ref{['mainthm']} and Corollary \ref{['maincor']}

Key Result

Theorem 1.1

Let $X$ be a canonical weak $\mathbb{Q}$-Fano $3$-fold. Then

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: cf. Positivity2
  • Definition 2.4
  • Lemma 2.5: cf. KM
  • Lemma 2.6: cf. Shokurov, Kol92
  • Definition 2.7
  • Lemma 2.8: cf. HLQ21, HJL21
  • ...and 16 more