The Complete Intersection property for binomial ideals of collections of cells

In this paper, we provide a combinatorial characterization of those collections of cells whose inner $2$-minor ideals are complete intersections. More precisely, given a collection of cells $\mathcal{C}$ and its associated inner $2$-minor ideal $I_{\mathcal{C}}$, we prove that $I_{\mathcal{C}}$ is a complete intersection if and only if $\mathcal{C}$ is a chessboard.