examples of toric varieties

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examples of toric varieties

Zero: \({\mathbb{G}}_m^n\)

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A full basis: \({\mathbb{A}}^n\)

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\({\mathbb{P}}^1\)

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\({\mathbb{P}}^2\)

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\({\mathbb{P}}^n\)

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\({\mathbb{P}}^1\times {\mathbb{P}}^1\)

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Weighted projective spaces

\({\mathbb{P}}(1,1,2)\)

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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 \(C_d\)

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Rational normal scrolls

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The Hirzebruch surface \(H_r\)

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Dual cones and faces

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Cone over a quadric

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Cone over the rational normal curve

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Misc

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

attachments/Pasted%20image%2020220621235637.png This yields a union of cones: attachments/Pasted%20image%2020220621235819.png

Blowups

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Examples of cones and fans

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Exercises

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