Admissible bases and generic lattice ideals

We introduce the notion of an admissible basis for rank-three positive lattices in $\mathbb{Z}^4$, namely a $\mathbb{Z}$-basis $\{{\bf c}_1,{\bf c}_2,{\bf c}_3\}$ satisfying three explicit sign conditions \textup{(I)}, \textup{(II)}, and \textup{(III)} on the coordinates of the basis vectors. We study lattice vectors of the form ${\bf u}^{(\mu,\lambda)}=\mu{\bf c}_1+{\bf c}_2+\lambda{\bf c}_3,$ where $\mu$ and $\lambda$ are positive integers. Under conditions \textup{(I)}, \textup{(II)}, and \textup{(III)}, we obtain a complete characterization of the vectors ${\bf u}^{(\mu,\lambda)}$ with positive first and fourth coordinates that are neighbors of the origin: such a vector is a neighbor if and only if $\mu=1$ and $\lambda\le2$. As an application, for every integer $m\ge1$ we construct a positive lattice $L(m)\subseteq \mathbb{Z}^4$ admitting an admissible basis whose associated lattice ideal is generic. This yields an infinite family of generic lattice ideals with exactly seven minimal binomial generators. This family shows that the characterization is sharp, since both ${\bf u}^{(1,1)}$ and ${\bf u}^{(1,2)}$ occur as neighbors of the origin.