Rigid Gorenstein toric Fano varieties arising from directed graphs

Rigid Gorenstein toric Fano varieties arising from directed graphs

A directed edge polytope \(\mathcal{A}_G\) is a lattice polytope arising from root system \(A_n\) and a finite directed graph \(G\). If every directed edge of \(G\) belongs to a directed cycle in \(G\), then \(\mathcal{A}_G\) is terminal and reflexive, that is, one can associate this polytope to a G...

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Veröffentlicht in:arXiv.org 2021-10
Hauptverfasser: Selvi, Kara, Portakal, Irem, Tsuchiya, Akiyoshi
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Graph theory
Graphs
Mathematics - Algebraic Geometry
Mathematics - Combinatorics
Polytopes
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description A directed edge polytope \(\mathcal{A}_G\) is a lattice polytope arising from root system \(A_n\) and a finite directed graph \(G\). If every directed edge of \(G\) belongs to a directed cycle in \(G\), then \(\mathcal{A}_G\) is terminal and reflexive, that is, one can associate this polytope to a Gorenstein toric Fano variety \(X_G\) with terminal singularities. It is shown by Totaro that a toric Fano variety which is smooth in codimension \(2\) and \(\mathbb{Q}\)-factorial in codimension \(3\) is rigid. In the present paper, we classify all directed graphs \(G\) such that \(X_G\) is a toric Fano variety which is smooth in codimension \(2\) and \(\mathbb{Q}\)-factorial in codimension \(3\).
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subjects Graph theory
Graphs
Mathematics - Algebraic Geometry
Mathematics - Combinatorics
Polytopes
title Rigid Gorenstein toric Fano varieties arising from directed graphs
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