Toric initial ideals of $\Delta$-normal configurations: Cohen-Macaulayness and degree bounds

O'Shea, Edwin ; Thomas, Rekha R.

Journal of Algebraic Combinatorics, Tome 21 (2005) no. 3, pp. 247-268.

Résumé

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

Détail

Keywords: key words toric ideals, triangulations, Hilbert bases, Gröbner bases