  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, non-prescribability of faces, and realizability of
combinatorial polytopes.

Combinatorics;Commutative Algebra;Algebraic Geometry