ScalingStacks

Example 4.32 . [02QA]

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Example 4.32.

We can use the above description of the restriction of a line bundle to an orbit to compute the degree of an orbit of dimension one. Let Σ\Sigma be a complete fan and τ∈Σn−1\tau\in\Sigma^{n-1}. Hence V⁡(τ)V(\tau) is a toric curve. Let σ1\sigma_{1} and σ2\sigma_{2} be the two nn-dimensional cones that have τ\tau as a common face. Let Ψ\Psi be a virtual support function. Choose v∈σ1v\in\sigma_{1} such that πτ​(v)\pi_{\tau}(v) is a generator of the lattice N⁡(τ)N(\tau). Then, by (4.25) and (4.30),

(4.33) degDΨ⁡(V⁡(τ))=deg⁡(ιτ∗​DΨ)=mσ2​(v)−mσ1​(v).\deg_{D_{\Psi}}(V(\tau))=\deg(\iota_{\tau}^{*}D_{\Psi})=m_{\sigma_{2}}(v)-m_{\sigma_{1}}(v).

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