A classification algorithm for reflexive simplices

TL;DR

The paper tackles the classification of reflexive simplices, equivalently Gorenstein fwps, in dimensions $5$ and $6$ by developing a degree-matrix, Gale-dual framework supplemented with an explicit automorphism group for invariant-factor abelian groups. It provides a constructive algorithm that translates reflexivity into concrete matrix conditions, enabling exhaustive enumeration (up to unimodular equivalence) and direct toric data extraction, including explicit formulas for the Picard group and Gorenstein index. A coherent suite of results connects weight vectors, torsion vectors, and degree matrices, culminating in an actionable normal-form based classification that yields $220{,}794$ and $309{,}019{,}970$ classes in dimensions $5$ and $6$, respectively. The work has significant implications for toric geometry and mirror-symmetry via explicit Fano polytope data and invariant computations, with practical runtimes demonstrated on multi-core hardware.

Abstract

We present a general classification algorithm for reflexive simplices, which allows us to determine all reflexive simplices in dimensions five and six. In terms of algebraic geometry this means that we classify the Gorenstein fake weighted projective spaces in dimensions five and six. As a byproduct of our methods, we obtain explicit formulae for the Picard group and the Gorenstein index of any fake weighted projective space.

Theorems & Definitions (37)

• Corollary 1.2
• Theorem 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.5
• proof
• Remark 2.6
• Example 2.7
• Proposition 2.8
• proof