An effective upper bound for Fano indices of canonical Fano threefolds, I

TL;DR

This work establishes a sharp upper bound $ ext{q}_{\mathbb{Q}}(X)\le 61$ for the $\mathbb{Q}$-Fano index of $ abla$-factorial weak Fano $3$-folds with isolated canonical singularities, by weaving together Reid's basket, a Kawamata--Miyaoka type inequality, and a tailored Riemann--Roch framework via sequential terminalization. It introduces a rank-two foliation analysis to handle two stubborn potential extremal cases and shows those lead to contradictions, thereby supporting a conjectured universal bound. The authors also provide a concrete computational strategy to search the invariant data space, and they develop tools to compute sections and indices through a Weil pullback RR formula. Overall, the paper advances canonical Fano geometry by tightening index bounds and refining the methodological toolkit for singular Fano threefolds, with potential extensions to broader singular settings.

Abstract

Let $X$ be a $\mathbb Q$-factorial weak Fano $3$-fold with at worst isolated canonical singularities. We show that the $\mathbb Q$-Fano index of $X$ is at most $61$.

Theorems & Definitions (48)

• Conjecture 1.1: cf. wang*Conjecture 3.7
• Example 1.2: kasprzyk2
• Theorem 1.3
• Lemma 2.1: cf. jiang-liu-liu*Lemma 5.1, Proposition 5.2
• proof
• Remark 2.2
• Theorem 2.3: jiang-liu-liu*Theorem 3.8
• proof
• Theorem 3.1: jiang-liu-liu*Theorem 4.2
• proof