On Gorenstein Fano toric complete intersections

TL;DR

The paper classifies $\mathbb{Q}$-factorial Gorenstein Fano general toric complete intersections of rank one inside fake weighted projective spaces, introducing a downgrading framework to bound torsion in divisor class groups and leveraging weight-degree constellations to enumerate all possible configurations. It proves three main theorems that enumerate true Gorenstein Fano weight-degree constellations for types $(3,1)$, $(3,2)$, and $(3,3)$, and then compiles complete lists of degree data and geometric invariants for every resulting gtci, totaling 78 families. The results provide explicit, computable data (degree matrices, anticanonical self-intersection, and global sections) for all cases and connect them to classical smooth gtci, yielding a comprehensive toric Fano classification framework. This advances the understanding of non-degenerate toric complete intersections in fwps and supplies a robust dataset for further algebraic and birational geometric investigations.

Abstract

We classify Q-factorial Gorenstein Fano non-degenerate complete intersection threefolds in fake weighted projective spaces.

Theorems & Definitions (61)

• Theorem 1.1
• Example 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Example 2.6
• Example 2.8
• Proposition 2.9
• proof
• Proposition 2.10