Toric weak Fano varieties associated to building sets (Q1626400)

One can construct nonsingular projective toric varieties from building sets. A building set on a nonempty finite set \(S\) is a finite set of nonempty subsets of \(S\) satisfying certain conditions. A nonsingular projective algebraic variety is called weak Fano if its anticanonical divisor is nef and big. The author characterizes building sets whose associated toric varietes are weak Fano. Moreover, a finite simple graph defines a building set. One can associate to a finite simple graph an integral convex polytope and thus a projective toric variety. In a previous paper of the author [``Toric Fano varieties associated to building sets'', Kyoto J. Math. 60, No. 1, 45--59 (2020; \url{doi:10.1215/21562261-2019-0034})], they prove that if the associated toric variety to a building set is Fano, then the polytope associated to the toric variety can be obtained from a finite directed graph. On the other hand in this paper, they present examples where the toric variety \(X\) associated to a certain building set is weak Fano but the associated reflexive polytope to \(X\) cannot be obtained from a graph. There also exists a reflexive polytope associated to a finite simple graph which cannot be obtained from a building set.