Contents

# Contents

## Idea

A kind of algebraic variety generalizing a torus with its abelian group 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 $\Delta_2$.

## References

• 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 16:33:49. See the history of this page for a list of all contributions to it.