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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 Gröbner basis generated in degree at most dim( $${I}_{\cal A}$$ ) ? In this paper, we answer both questions positively for 驴-normal configurations. These are normal configurations that admit a regular triangulation 驴 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 驴-normal and non-驴-normal configurations.

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