Toric Initial Ideals of Δ-Normal Configurations: Cohen-Macaulayness and Degree Bounds

2003/08/12 by Edwin O'Shea, O'Shea, Edwin, Rekha R. Thomas +1
Mathematics · #05E02 #13P02 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.CO #msc:05E02 #msc:13P02

paper · pdf · doi:10.48550/arxiv.math/0308109

18 pages, 2 figures

arxiv created 2003/08/12 · arxiv updated 2009/12/01

Abstract

A normal (respectively, graded normal) vector configuration A defines the toric ideal I_A of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when A is normal and graded, I_A is generated in degree at most the dimension of I_A. Based on this, Sturmfels asked if these properties extend to initial ideals -- when A is normal, is there an initial ideal of I_A that is Cohen-Macaulay, and when A is normal and graded, does I_A have a Gröbner basis generated in degree at most dim(I_A) ? In this paper, we answer both questions positively for Δ-normal configurations. These are normal configurations that admit a regular triangulation Δ with the property that the subconfiguration in each cell of the triangulation is again normal. Such configurations properly contain among them all vector configurations that admit a regular unimodular triangulation. We construct non-trivial families of both Δ-normal and non-Δ-normal configurations.

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