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

Abstract

A normal (respectively, graded normal) vector configuration $${\cal A}$$ defines the toric ideal $${I}_{\cal A}$$ of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when $${\cal A}$$ is normal and graded, $${I}_{\cal A}$$ is generated in degree at most the dimension of $${I}_{\cal A}$$ . Based on this, Sturmfels asked if these properties extend to initial ideals--when $${\cal A}$$ is normal, is there an initial ideal of $${I}_{\cal A}$$ that is Cohen-Macaulay, and when $${\cal A}$$ is normal and graded, does $${I}_{\cal A}$$ have a Grobner basis generated in degree at most dim( $${I}_{\cal A}$$ ) ? In this paper, we answer both questions positively for ?-normal configurations. These are normal configurations that admit a regular triangulation ? 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 ?-normal and non-?-normal configurations.