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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
