Toric varieties of Schr\"{o}der type

Abstract

A dissection of a polygon is obtained by drawing diagonals such that no two diagonals intersect in their interiors. In this paper, we define a toric variety of Schr\"{o}der type as a smooth toric variety associated with a polygon dissection. Toric varieties of Schr\"{o}der type are Fano generalized Bott manifolds, and they are isomorphic if and only if the associated Schr\"{o}der trees are the same as unordered rooted trees. We describe the cohomology ring of a toric variety of Schr\"{o}der type using the associated Schr\"{o}der tree and discuss the cohomological rigidity problem.