GROBNER BASES OF TORIC VARIETIES

Abstract

In this article projective toric varieties are studied from the viewpoint of Gröbner basis theory and combinatorics. We characterize the radicals of all initial ideals of a toric variety X{\mathscr A} as the Stanley-Reisner ideals of regular triangulations of its set of weights {\mathscr A}. This implies that the secondary polytope Σ({\mathscr A}) is a Minkowski summand of the state polytope of X{\mathscr A}. Here the lexicographic (resp. reverse lexicographic) initial ideals of X{\mathscr A} arise from triangulations by placing (resp. pulling) vertices. We also prove that the state polytope of the Segre embedding of P^r-1×P^s-1 equals the secondary polytope Σ(Δ_r-1×Δ_s-1) of a product of simplices.