Regular subdivisions, bounds on initial ideals, and categorical limits
George Balla, Daniel Corey, Igor Makhlin, Victoria Schleis
TL;DR
The paper develops a general framework linking initial ideals to regular subdivisions via a point configuration $\mathcal{A}(I)$ associated to any homogeneous ideal $I$. It defines concrete lower and upper bounds $I_w$ and $I^w$ on $\mathrm{in}_w I$ from the subdivision data of $\mathcal{A}(I)$ and interprets these bounds categorically as colimits and limits over face posets, with $R_w$ and $R^w$ respectively. It proves that the sets of weights where these bounds are exact form subfans of the secondary fan (and its reflection), and shows how these results extend to very affine schemes and to Grassmannians, including new families of smooth initial degenerations. The work also provides a practical algorithm to compute the associated point configuration $\mathcal{A}(I)$ and the bounds using the OSCAR package, unifying toric, Grassmannian, and matroid subdivision phenomena under a single framework with explicit (co)limit descriptions. Altogether, the results give a systematic, category-theoretic lens on when and how initial degenerations can be controlled by regular subdivisions across broad classes of projective schemes.
Abstract
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 $A$.