Polycubes and polycube ideals
To a polycube $\mathcal P$ in $\mathbb R^n$, we associate an ideal $I_{\mathcal P}$ generated by the join--meet relations of vertices that are corners of a cuboid of $\mathcal P$. For $n=2$ these are the polyomino ideals. In contrast with the planar case, we show that for $n\geq 3$ the ideal $I_{\mathcal P}$ is prime if and only if $\mathcal P$ is a cuboid. For convex polycubes in $\mathbb R^3$ of uniform thickness whose vertex set is a sublattice of $\mathbb N^3$, the defining relations form a quadratic Gröbner basis, and we characterize the Cohen--Macaulay ones in terms of a poset on the vertices of $\mathcal P$; in this case the order complex of the poset is vertex decomposable. Finally, as an application, we show that the polycube algebra of a thickened $L$-convex polyomino is Koszul, and we characterize when it is Cohen--Macaulay.