Vanishing ideals over complete multipartite graphs

We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\text{G}={\mathcal{K}}_{{\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_r}$ 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 $\text{G}$ and which generate the \emph{toric algebra of $\text{G}$} (type II generators) and a set of binomials of degree $q-1$ obtained combinatorially from $\text{G}$ (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo–Mumford regularity is equal to $\max \{{\alpha }_1(q-2),\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_r(q-2),\lceil (n-1)(q-2)\mathrm{/}2\rceil \}$, where $n={\alpha }_1+\mathrm{.}\mathrm{.}\mathrm{.}+{\alpha }_r$.