Vanishing ideals over complete multipartite graphs

We study the vanishing ideal of the parametrized algebraic toric set associated to the complete multipartite graph G=Kα1,…,αrG=Kα1,…,αr over a finite field of order q  . We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in GG and which generate the toric algebra of  GG (type II generators) and a set of binomials of degree q−1q−1 obtained combinatorially from GG (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo–Mumford regularity is equal to max{α1(q−2),…,αr(q−2),⌈(n−1)(q−2)/2⌉}max{α1(q−2),…,αr(q−2),⌈(n−1)(q−2)/2⌉}, where n=α1+⋯+αrn=α1+⋯+αr.