Hypertoric varieties and zonotopal tilings

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.