Regular subdivisions, bounds on initial ideals, and categorical limitsWe extend two known constructions that relate regular subdivisions to initial degenerations of projective toric varieties and Grassmannians. We associate a point configuration $A$ with any homogeneous ideal $I$. We obtain upper and lower bounds on each initial ideal of $I$ in terms of regular subdivisions of $A$. Both bounds can be interpreted categorically via limits over face posets of subdivisions. We also investigate when these bounds are exact. This is an extended abstract of a forthcoming paper.
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