ncts

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 gamma vector equal to 0. The gamma vector is an invariant of palindromic polynomials whose nonnegativity lies between unimodality and real-rootedness. It is expected that the cases where the gamma vector is 0 form "building blocks'' among 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 for such toric varieties, we find that the interpretation of the top component of the gamma vector as a signature encodes *intrinsic* combinatorial information on the fan in addition to compatibility with existing natural combinatorial examples as shown earlier. Our main method involves 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. 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 expected to cover the minimal objects described above. Note that this involves intersections of rational equivalence relations with 2-dimensional orbit closures instead of 1-dimensional ones as in most combinatorial applications.