Degrees of non-Gorenstein canonical Fano threefolds with Picard number one

TL;DR

The article determines the sharp upper bound $c_1(X)^3 \le \tfrac{200}{3}$ for the anticanonical degree of non-Gorenstein $\mathbb{Q}$-factorial canonical Fano threefolds with Picard number one, by a comprehensive case analysis stratified by the $\mathbb{Q}$-Fano index $q_{\mathbb{Q}}(X)$, the Weil index $q_{\mathbb{W}}(X)$, and the local index data $J_A$. It combines Reid basket data, Kawamata-Miyaoka-type inequalities, and divisibility constraints to prune possibilities, organizing the work into $q\le 5$, $q=6$, and $q\ge 7$ regimes and further subdividing by torsion and index relations. The bound is shown to be tight, realized by the toric threefold $X\cong \mathbb{P}(1,1,3,5)$ with $c_1(X)^3=\tfrac{200}{3}$ and $\mathrm{q}_{\mathbb{W}}(X)=10$, with equality forcing $\mathrm{q}_{\mathbb{Q}}(X)=\mathrm{q}_{\mathbb{W}}(X)\in\{2,4,5,10,20,40\}$. Overall, the work advances the classification of high-degree non-Gorenstein canonical Fano threefolds and highlights the interplay between singularities, index theory, and birational reductions via the MMP.

Abstract

We show that the optimal upper bound of the anticanonical degrees of non-Gorenstein $\mathbb{Q}$-factorial canonical Fano threefolds with Picard number one is 200/3.

Theorems & Definitions (24)

• Theorem 1.1: Corollary 4.3
• Example 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3: reid87*(10.3), jll25*Theorem 2.1
• Lemma 2.4
• proof
• Theorem 2.5: jll25*Theorem 4.2
• Definition 2.6
• Lemma 2.7