Hypertoric varieties and zonotopal tilings

TL;DR

This paper introduces a new, abstract framework for hypertoric varieties, associating them with zonotopal tilings, including irregular ones, and conjectures a complete classification akin to toric varieties.

Contribution

It generalizes the definition of hypertoric varieties and constructs new examples from irregular tilings, proposing a comprehensive classification scheme.

Findings

Constructed hypertoric varieties from any zonotopal tiling.

Identified that known examples correspond to regular tilings.

Proposed conjecture for a complete classification of hypertoric varieties.

Abstract

We give an abstract definition of a hypertoric variety, generalizing the existing constructive definition. We construct a hypertoric variety associated with any zonotopal tiling, and we show that the previously known examples are exactly those varieties associated with regular tilings. In particular, the examples that we construct from irregular tilings have not appeared before. We conjecture that our construction gives a complete classification of hypertoric varieties, analogous to the classification of toric varieties by fans.