Vanishing ideals over complete multipartite graphs

We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{\alpha_1,...,\alpha_r}$ 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 \emph{toric algebra of $\G$} (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 $\max\set{\alpha_1(q-2),...,\alpha_r(q-2), \lceil (n-1)(q-2)/2\rceil}$, where $n=\alpha_1+... + \alpha_r$.