The web of reflexive polygons is connected

TL;DR

This paper explores the connectedness of webs of lattice polytopes, especially reflexive polygons, using toric Mori theory and combinatorial descriptions, proving their connectivity through inclusion relations.

Contribution

It introduces a geometric approach to the connectedness problem and provides a combinatorial description of toric Sarkisov links in terms of primitive generating sets.

Findings

Reflexive polygons form a single connected web via inclusion relations.

The connectedness is established both constructively and non-constructively.

The approach extends to terminal polygons and their inclusion relations.

Abstract

We discuss the problem on the connectedness of various webs of lattice polytopes by introducing a geometric point of view from the toric Mori theory. To this end, we provide a combinatorial description of toric Sarkisov links in terms of certain sets of lattice points, which we call primitive generating sets. In two dimensions, the description is further translated into the language of lattice polygons. As an application, we prove in two ways (constructive and non-constructive) that reflexive or terminal polygons form a single connected web via inclusion relations even without taking modulo unimodular equivalences.