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examples of toric varieties

Zero: ${\mathbf{G}}_m^n$

A full basis: ${\mathbf{A}}^n$

${\mathbf{P}}^1$

${\mathbf{P}}^2$

${\mathbf{P}}^n$

${\mathbf{P}}^1\times {\mathbf{P}}^1$

Weighted projective spaces

${\mathbf{P}}(1,1,2)$

Cuspidal curves

The cuspidal curve $V\left(x^{3}-y^{2}\right) \cong\left\{\left(t^{2}, t^{3}\right) \mathrel{\Big|}t \in \mathbb{C}^{2} \subseteq \mathbb{A}^{2}\right.$ is an affine toric variety: $$\begin{align*} \mathbb{C}^{*}=\left\{\left(t^{2}, t^{3}\right) \mathrel{\Big|}t \neq 0\right\} \subseteq V\left(x^{3}-y^{2}\right) \end{align*}$$

Rational normal curves $C_d$

Rational normal scrolls

The Hirzebruch surface $H_r$

Dual cones and faces

Cone over a quadric

Cone over the rational normal curve

Misc

$\operatorname{Spec}{\mathbf{C}}[x,y,z]/\left\langle{x^2-yz}\right\rangle$

This yields a union of cones:

Blowups

Examples of cones and fans

Exercises