Binomial ideals attached to finite collections of cells (Q6600736)

The authors are considering ideals of inner \(2\)-minors, attached to finite collections of cells. Let \(P\) be a polyomino and let \(S=K[\{x_{\mathbf{a}}\}_{\mathbf{a}\in V(P)}]\) be the polynomial ring in \(\vert V(P)\vert\) variables over \(K\). Let \(I_P\) be the binomial ideal generated by the 2-minors of \(P\). The first rersult is that if \(P\) is a finite set of \(c\) cells, then \(height(I_P)\leq c\leq bigheight(I_P)\). The second result is that if \(K\) is a perfect field, \(P\) is a finite set of cells such that \(I_P\) is a prime ideal, then \(S/I_p\) has an isolated singularity if and only if \(P\) is an inner interval.