ScalingStacks

Remark 8.27 . [02YP]

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Remark 8.27.

We check A1,1​(0,1)=A1,1​(1,0)=3/4A_{1,1}(0,1)=A_{1,1}(1,0)={3}/{4}. Let b≥0b\geq 0 and let 𝒪𝔽b​(1)¯{\overline{{\mathcal{O}}_{\mathbb{F}_{b}}(1)}} the adelic line bundle on 𝔽b\mathbb{F}_{b} associated to a0=1a_{0}=1 and a1=b+1a_{1}=b+1. Putting n=r=1n=r=1, a0=1a_{0}=1 and a1=b+1a_{1}=b+1 in Proposition 8.26, we recover the expression for the height of Hirzebruch surfaces established in [Mou06]: h𝒪𝔽b​(1)¯⁡(𝔽b)=12​b2+94​b+3\operatorname{h}_{{\overline{{\mathcal{O}}_{\mathbb{F}_{b}}(1)}}}(\mathbb{F}_{b})=\frac{1}{2}b^{2}+\frac{9}{4}b+3.

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