Weighted projective spaces and reflexive simplices
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.
scientific article; zbMATH DE number 1747286