In this paper we introduce a natural model for the realization space of a polytope up to projective equivalence which we call the slack realization space of the polytope. The model arises from the positive part of an algebraic variety determined by the slack ideal of the polytope. This is a saturated determinantal ideal that encodes the combinatorics of the polytope. We also derive a new model of the realization space of a polytope from the positive part of the variety of a related ideal. The slack ideal offers an effective computational framework for several classical questions about polytopes such as rational realizability, nonprescribability of faces, and realizability of combinatorial polytopes.

We study the partially ordered set $P(a_1,\ldots, a_n)$ of all multidegrees $(b_1,\dots,b_n)$ of monomials $x_1^{b_1}\cdots x_n^{b_n}$, which properly divide $x_1^{a_1}\cdots x_n^{a_n}$. We prove that the order complex $\Delta(P(a_1,\dots,a_n))$ of $P(a_1,\ldots a_n)$ is (nonpure) shellable by showing that the order dual of $P(a_1,\ldots,a_n)$ is $CL$-shellable. Along the way, we exhibit the poset $P(4,4)$ as a new example of a poset with $CL$-shellable order dual that is not $CL$-shellable itself. For $n = 2$, we provide the rank of all homology groups of the order complex $\Delta ( P(a_1,a_2) )$. Furthermore, we give a succinct formula for the Euler characteristic of $\Delta ( P(a_1,a_2) )$.