Toric varieties of Schr\"{o}der type

TL;DR

This paper introduces a class of smooth toric varieties linked to polygon dissections, explores their properties as Fano generalized Bott manifolds, and investigates their cohomology and rigidity based on associated Schr"{o}der trees.

Contribution

It defines toric varieties of Schr"{o}der type, characterizes their isomorphism conditions, and describes their cohomology rings using Schr"{o}der trees.

Findings

Toric varieties of Schr"{o}der type are Fano generalized Bott manifolds.

Isomorphism classes correspond to unordered Schr"{o}der trees.

Cohomology rings are described via associated Schr"{o}der trees.

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.