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

Edwin O'Shea·Rekha R. Thomas
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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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Edwin O'Shea, Rekha R. Thomas. 2003-08-12. Toric Initial Ideals of $Δ$-Normal Configurations: Cohen-Macaulayness and Degree Bounds. https://arxiv.org/abs/math/0308109

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