Algebraic properties of some configurations of fat lines

In this paper we analyze the algebraic properties of some configurations of fat lines of $${\mathbb {P}}^3_K$$ , with different multiplicity conditions on the lines, called $$(m_1, m_2, m_3)$$ -fat complete grids of type $$(\ell _1,\ell _2,\ell _3)$$ . We study the connection of these grids of lines with liftings of monomial ideals, we explicitly describe the generators of the ideal defining these grids, we study the Cohen-Macaulay property and the syzygy modules of these configurations of fat lines.