Weighted projective spaces and reflexive simplices
type: Item
instance of: scholarly article
title: Weighted projective spaces and reflexive simplices
zbMATH Open document ID: 1013.52009
author: Heinke Conrads
published in: Manuscripta Mathematica
publication date: 2002-05-29T00:00:00Z
review text: An integral simplex is called reflexive if its dual is again integral. Up to unimodular equivalence, the number of reflexive polytopes is known to be finite for every dimension \(n\), but a classification is only known for \(n\leq 4\). The author provides an algorithm which classifies reflexive simplices in arbitrary dimension \(n\). To achieve this classification, he first defines a type \((Q_{\text{red}},\lambda)\) of a \(\mathbb{Z}^n\)-integral simplex \(\Delta= (v_0,\dots, v_n)\) which consists of the reduced weight \({1\over\lambda} (q_0,\dots, q_n)\), where \(q_i=|\det(v_0, \dots,\widehat v_i,\dots,v_n) |\), and a factor \(\lambda=\text{gcd}(q_0,\dots, q_n)\). Using weighted projective spaces, he shows how one can construct a simplex of type \((Q_{\text{red}},1)\) for every \(Q\in \mathbb{N}^{n+1}\). Then he shows that, for any given pair \((Q_{\text{red}}, \lambda)\), there exist only finitely many simplices of this type, and they can all be obtained as images of the simplex of type \((Q_{\text{red}},1)\) already constructed. Since the possible factors \(\lambda\) can be computed effectively, the author obtains a classification algorithm for reflexive simplices of dimension \(n\). An implementation yields for \(n\geq 4\) a list which coincides with the previously known classification.
reviewed by: Martin Kreuzer
Mathematics Subject Classification ID: 52B20
Mathematics Subject Classification ID: 52B12
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 1747286
zbMATH Keywords: integral polytope
zbMATH Keywords: reduced weight
zbMATH Keywords: weighted projective space
zbMATH Keywords: reflexive simplices
MaRDI profile type: Publication
full work available at URL: s002290100235
OpenAlex ID: W2050266449
DOI: 10.1007/S002290100235
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description: scientific article; zbMATH DE number 1747286