nLab toric variety




A kind of algebraic variety generalizing a torus with its abelian group structure.

Combinatorial Aspects

A fan Δ\Delta is a collection of cones closed under the operations of taking faces and intersections. Each cone gives rise to an affine variety. The result of gluing these along intersections gives the toric variety of this fan X ΔX_\Delta.

This correspondence extends functorially. Fan morphisms between a fan Δ 1\Delta_1 in N 1N_1 to Δ 2\Delta_2 in N 2N_2 is a linear map ff from N 1N_1 to N 2N_2 such that every cone σΔ 1\sigma \in \Delta_1 goes to f(σ)σ 2f (\sigma) \subset \sigma_2 where σ 2\sigma_2 is a cone in Δ 2\Delta_2.

Orbit-Cone Correspondence


  • Ezra Miller, What is… a toric variety?, Notices of the AMS, volume 55, number 5 (pdf)

  • Pavel Dimitrov, Toric varieties, a short introduction (pdf)

  • Stephan Fischli, On Toric Varieties (pdf)

  • Helena Verrill, David Joyner, Notes on toric varieties (2002) (pdf)

Last revised on August 14, 2019 at 20:33:49. See the history of this page for a list of all contributions to it.