Quadratic Gröbner bases for smooth 3×3 transportation polytopes

Abstract

Abstract. The toric ideals of 3 × 3 transportation polytopes T rc are quadratically generated. The only exception is the Birkhoff polytope B 3 .If T rc is not a multiple of B 3 , these ideals even have squarefree quadratic initial ideals. This class contains all smooth 3 × 3 transportation polytopes.1. IntroductionIf X P is smooth (the normal fan of P is unimodular), then L P is very ample, and provides an embedding X P ֒→ P r−1 , where r = #(P ∩ d ). So we can think of X P as canonically sitting in projective space. The following question [Stu97, Conjecture 2.9] about the defining equations of X P ⊂ P r−1 has been around for quite a while, but its origins are hard to track (cf. [BCF + 05]).Question. Let P be a lattice polytope whose corresponding projective toric variety is smooth. Is the defining ideal I P generated by quadratics?There are two variations of this question (which are of strictly increasing strength).• Is the homogeneous coordinate ring [x 1 , . . . , x r ]/I P Koszul?• Does I P have a quadratic Gröbner basis? The last version has a combinatorial interpretation. It asks for the existence of very special, "quadratic" triangulations of P , see §1.3 below.1.2. Results. Simple transportation polytopes provide a large family of smooth polytopes. Yet, the 3 × 3 Birkhoff polytope B 3 is a nonsimple transportation polytope whose ideal is not generated by quadratic polynomials. In this note, we show that in the 3 × 3 case, this is the only example. In Section 2, we show that B 3 is the only (3 × 3)-transportation polytope whose ideal is not quadratically generated.

Diagonal splittings of toric varieties and unimodularity

“…However, as noted by Haase and Paffenholz [HP09], for n = 3, the polytope B 3 corresponds to an embedding of X(Σ 3 ) as the cubic hypersurface x 0 x 1 x 2 = x 3 x 4 x 5 in the projective space P 5 . Since the homogeneous ideal of this embedding is not generated by quadrics, it follows that X × X × X has no splitting compatible with (∆ × X) ∪ (X × ∆).…”

On the equations and classification of toric quiver varieties

“…More importantly to each projective toric quiver variety there are an infinite number of quiver polytopes associated, hence there did not seem to be any obvious way to achieve our goal via direct computation. Instead we follow an approach similar to that of [16], where it was shown that amongst the 3 × 3 transportation polytopes, only the Birkhoff polytope B 3 yields a toric ideal which is not generated in degree two. This is a special case of our result, since 3 × 3 transportation polytopes are quiver polytopes of the bipartite quiver K 3,3 .…”

“…The aim of our work in Section 4 is to refine this result by listing the quiver polytopes up to a fixed dimension that yield a toric ideal that can not be generated in degree two. One of the key tools we use here is a hyperplane subdivision method which was also applied in [16] to study the toric ideals of 3 × 3 transportation polytopes. This method allows us to estimate the generators of the toric ideals of quiver polytopes by finding generators for some small subpolytopes which we call quiver cells.…”

“…(i) The above argument differs from Section 2 of[16] which depends on the fact that in the case of 3 × 3 transportation polytopes the cells that do not yield cubic relations, also possess a quadratic Gröbner basis (in fact they are simplices). (ii) An alternative proof of Lemma 4.6 could be derived from Theorem 6.2 in[7], which is proven using several facts about the defining ideals of monoidal complexes.…”

Bounds on the coefficients of tension and flow polynomials

“…One important property of this method of refinement is that pulling a vertex preserves regularity of the subdivision: If C is a regular subdivision of a polytope P and C is a pulling refinement of C, then C is a regular subdivision of P . See [14,20]. A lattice polytope P such that all pulling triangulations of P are unimodular is called compressed.…”