nLab toric variety

Idea

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

Combinatorial Aspects

A fan $\Delta$ 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 $X_\Delta$.

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

Orbit-Cone Correspondence

References

• Garth Warner: Abelian Theory, University of Washington [arXiv:2012.15736, pdf]

• Ezra Miller, What is… a toric variety?, Notices of the AMS 55 5 (2008) [pdf, full issue: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)