Regular subdivisions, bounds on initial ideals, and categorical limits

Several known constructions relate initial degenerations of projective toric varieties and Grassmannians to regular subdivisions of appropriate point configurations. We define a general framework which allows for partial generalizations of these constructions to arbitrary projective schemes (as well as their very affine parts). We associate a point configuration $A$ with any homogeneous ideal $I$. We obtain upper and lower bounds on every initial ideal of $I$, defining them in terms of the regular subdivision of $A$ given by the same weight. Furthermore, both bounds are interpreted categorically via (co)limits over the face poset of the subdivision. We also investigate when these bounds are exact, showing that the respective weights form a subfan in the secondary fan of $A$.