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Regular Unimodular Triangulations of Reflexive IDP 2-Supported Weighted Projective Space Simplices


For each integer partition \(\mathbf {q}\) with d parts, we denote by \(\Delta _{(1,\mathbf {q})}\) the lattice simplex obtained as the convex hull in \(\mathbb {R}^d\) of the standard basis vectors along with the vector \(-\mathbf {q}\). For \(\mathbf {q}\) with two distinct parts such that \(\Delta _{(1,\mathbf {q})}\) is reflexive and has the integer decomposition property, we establish a characterization of the lattice points contained in \(\Delta _{(1,\mathbf {q})}\). We then construct a Gröbner basis with a squarefree initial ideal of the toric ideal defined by these simplices. This establishes the existence of a regular unimodular triangulation for reflexive 2-supported \(\Delta _{(1,\mathbf {q})}\) having the integer decomposition property. As a corollary, we obtain a new proof that these simplices have unimodal Ehrhart \(h^*\)-vectors.

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Correspondence to Benjamin Braun.

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BB was partially supported by NSF award DMS-1953785. DH was partially supported by NSF award DUE-1356253.

Communicated by Liam Solus

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Braun, B., Hanely, D. Regular Unimodular Triangulations of Reflexive IDP 2-Supported Weighted Projective Space Simplices. Ann. Comb. (2021).

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Mathematics Subject Classification

  • Primary 52B20