Toric variety

Summary

A toric variety is a variety that contains an algebraic torus as a dense open subset. The action of the torus on itself must extend to the whole variety. Normal toric varieties correspond to combinatorial objects called fans, built from rational polyhedral cones. So geometric questions about them turn into questions about cones and polytopes, such as Newton polytopes. They are a standard testing ground for conjectures and constructions in algebraic geometry.