Toric Initial Ideals of Δ -Normal Configurations: Cohen-Macaulayness and Degree Bounds
Edwin O'Shea
and Rekha R. Thomas
Department of Mathematics University of Washington Seattle WA 98195-4350
DOI: 10.1007/s10801-005-6910-4
Abstract
A normal (respectively, graded normal) vector configuration A {\cal A} defines the toric ideal I A {I}_{\cal A} 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}_{\cal A} is generated in degree at most the dimension of I A {I}_{\cal A} . 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}_{\cal A} that is Cohen-Macaulay, and when A {\cal A} is normal and graded, does I A {I}_{\cal A} have a Gröbner basis generated in degree at most dim( I A {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.
-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.Pages: 247–268
Keywords: key words toric ideals; triangulations; Hilbert bases; Gröbner bases
Full Text: PDF
References
1. W. Bruns, J. Gubeladze, and N.V. Trung, “Normal polytopes, triangulations, and Koszul algebras,” J. Reine Angew. Math. 485 (1997), 123-160.
2. W. Bruns, J. Gubeladze, and N.V. Trung, “Problems and algorithms for affine semigroups,” Semigroup Forum 64(2) (2002), 180-212.
3. W. Bruns and R. Koch, “Computing the integral closure of an affine semigroup, Effective methods in algebraic and analytic geometry, 2000 (Krakow),” Univ. Iagel. Acta Math. 39 (2001), 59-70. Software: Normaliz, ftp.mathematik.Uni-Osnabrueck.DE/pub/osm/kommalg/software.
4. R.T. Firla, private correspondence.
5. R.T. Firla and G.M. Ziegler, “Hilbert bases, unimodular triangulations, and binary covers of rational polyhedral cones,” Discrete Comp. Geom. 21 (1999), 205-216.
6. A. Jensen, “CaTS, a software package for computing state polytopes of toric ideals,” available from
2. W. Bruns, J. Gubeladze, and N.V. Trung, “Problems and algorithms for affine semigroups,” Semigroup Forum 64(2) (2002), 180-212.
3. W. Bruns and R. Koch, “Computing the integral closure of an affine semigroup, Effective methods in algebraic and analytic geometry, 2000 (Krakow),” Univ. Iagel. Acta Math. 39 (2001), 59-70. Software: Normaliz, ftp.mathematik.Uni-Osnabrueck.DE/pub/osm/kommalg/software.
4. R.T. Firla, private correspondence.
5. R.T. Firla and G.M. Ziegler, “Hilbert bases, unimodular triangulations, and binary covers of rational polyhedral cones,” Discrete Comp. Geom. 21 (1999), 205-216.
6. A. Jensen, “CaTS, a software package for computing state polytopes of toric ideals,” available from