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

Original · EN

A normal (respectively, graded normal) vector configuration A defines the toric ideal Iₐ of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when A is normal and graded, Iₐ is generated in degree at most the dimension of Iₐ. Based on this, Sturmfels asked if these properties extend to initial ideals -- when A is normal, is there an initial ideal of Iₐ that is Cohen-Macaulay, and when A is normal and graded, does Iₐ have a Gröbner basis generated in degree at most dim(Iₐ)? 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.