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

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 Gr\"obner basis generated in degree at most $dim(I_A)$ ? In this paper, we answer both questions positively for $\Delta$-normal configurations. These are normal configurations that admit a regular triangulation $\Delta$ 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 $\Delta$-normal and non-$\Delta$-normal configurations.