Signature 0 toric varieties, wall crossings, and cross polytope-like structures

Abstract

We describe the structure of simplicial locally convex fans associated to even-dimensional complete toric varieties with signature 0. They belong to the set of such toric varieties whose even degree Betti numbers yield a top gamma vector component equal to 0. The gamma vector is an invariant of palindromic polynomials whose nonnegativity lies between unimodality and real-rootedness. It is known (and expected more generally) that the cases where this top component is 0 are among the "building blocks" of those where it is nonnegative. This means minimality with respect to a certain restricted class of blowups. However, this equality to 0 case is currently poorly understood. In the course of addressing this situation, we find that this interpretation encodes *intrinsic* combinatorial information on the fan in addition to earlier compatibility with existing natural combinatorial examples. Our main method uses wall crossings. The links of the fan come from a repeated suspension of the maximal linear subspace in its realization in the ambient space of the fan. Conversely, the centers of these links containing any particular line form a cone or a repeated suspension of one. The intersection patterns between these "anchoring" linear subspaces come from how far certain submodularity inequalities are from equality and parity conditions on their dimensions. This involves linear dependence and containment relations between them which are connected to optimization. We obtain these relations by viewing the vanishing of certain mixed volumes from the perspective of the exponents. Finally, these wall crossings yield a simple method of generating induced 4-cycles covering the minimal objects described above. We intersect rational equivalence relations with 2-dimensional orbit closures instead of 1-dimensional ones as in most combinatorial applications.