Vanishing ideals over complete multipartite graphs

Abstract

We study the vanishing ideal of the parametrized algebraic toric set associated to the complete multipartite graph G=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 G and which generate the toric algebra ofG (type II generators) and a set of binomials of degree q−1 obtained combinatorially from G (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⌉}, where n=α1+⋯+αr.